The Nature of X-Rays and Their Interactions with Matter 3031207432, 9783031207433

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Springer Tracts in Modern Physics 288

Joachim Stöhr

The Nature of X-Rays and Their Interactions with Matter

Springer Tracts in Modern Physics Volume 288

Series Editors Mishkatul Bhattacharya, Rochester Institute of Technology, Rochester, NY, USA Yan Chen, Department of Physics, Fudan University, Shanghai, China Atsushi Fujimori, Department of Physics, University of Tokyo, Tokyo, Japan Mathias Getzlaff, Institute of Applied Physics, University of Düsseldorf, Düsseldorf, Nordrhein-Westfalen, Germany Thomas Mannel, Emmy Noether Campus, Universität Siegen, Siegen, Nordrhein-Westfalen, Germany Eduardo Mucciolo, Department of Physics, University of Central Florida, Orlando, FL, USA William C. Stwalley, Department of Physics, University of Connecticut, Storrs, USA Jianke Yang, Department of Mathematics and Statistics, University of Vermont, Burlington, VT, USA

Springer Tracts in Modern Physics provides comprehensive and critical reviews of topics of current interest in physics. The following fields are emphasized: – – – –

Particle and Nuclear Physics Condensed Matter Physics Light Matter Interaction Atomic and Molecular Physics

Suitable reviews of other fields can also be accepted. The Editors encourage prospective authors to correspond with them in advance of submitting a manuscript. For reviews of topics belonging to the above mentioned fields, they should address the responsible Editor as listed in “Contact the Editors”.

Joachim Stöhr

The Nature of X-Rays and Their Interactions with Matter

Joachim Stöhr SLAC National Accelerator Laboratory Menlo Park, CA, USA

ISSN 0081-3869 ISSN 1615-0430 (electronic) Springer Tracts in Modern Physics ISBN 978-3-031-20743-3 ISBN 978-3-031-20744-0 (eBook) https://doi.org/10.1007/978-3-031-20744-0 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To Linda, My ski buddy, my best friend, my muse, my love, my wife.

Preface

Starting in the mid-1970s, I have had the extraordinary luck of experiencing and participating in two revolutions in x-ray science. It is the wonderful developments I witnessed during these nearly fifty years and my role in the development of x-ray techniques and the formulation of scientific cases for advanced synchrotron radiation (SR) facilities and x-ray free electron lasers (XFELs) that have led me to take on the endeavor of writing the present book. The other impetus was the recognition that the very process would allow me to learn what I craved to understand on a deeper level or for the first time. The period 1970–2000 saw the development of SR sources and science, based on the utilization of electron storage rings for the production of x-rays and the development of modern x-ray techniques. Storage ring-based sources offer high photon flux over a broad spectral range and by use of monochromators provided tunable highintensity x-rays for the first time, soon outdating previous x-ray sources based on Bremsstrahlung and characteristic line emission. The early development and use of SR continued in the 1980s–90s through the extension to undulator sources and optimization of dedicated electron storage rings. Today, the development of SR sources continues with the goal of building “ultimate” storage rings that produce diffraction limited x-ray beams into the multi-keV range. In the period 2005–2010, a second revolution occurred with the advent of XFELs based on the development of high brightness electron sources in conjunction with linear accelerators. In electron storage rings, the size of the continually circulating electron bunches is limited by an equilibrium between the forces of the focusing magnets that overcome the Coulombic repulsion between the electrons and the destabilizing radiation losses. Linear accelerators offer the advantage that the electron beam is not subjected to radiation losses, and exquisitely prepared electron bunches can therefore be used to create radiation in a long undulator, with the electron bunch then being discarded. While the repetition rate of linacs is considerably lower than that of storage rings, each XFEL pulse is in principle sufficient to obtain a spectrum or diffraction pattern. Over the years, my particular interest and efforts have been the development of soft x-ray techniques and their applications. My first book NEXAFS Spectroscopy, vii

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published in 1992 [1], dealt with the use of x-rays for problems in surface science and chemistry. It was influenced by what I learned at EXXON Research and Engineering Company about catalytic processes and about thin polymer films at the IBM Almaden Research Center. My second book, Magnetism: From Fundamentals to Nanoscale Dynamics, co-authored with H. C. Siegmann and published in 2006 [2] was largely the result of what I learned from my colleagues at IBM Almaden, from the literature and especially from Hans Siegmann. By inviting Hans to Stanford in 2000 after he had to retire from ETH, I helped him to actively continue what he loved most, doing physics. He taught me the mysteries and magic of magnetism and in the process allowed me to tap into his deep knowledge of science. The present book was written during the last ten years or so, after finishing management periods as Director of the Stanford Synchrotron Radiation Lightsource (SSRL) and the Linac Coherent Light Source (LCLS) at SLAC/Stanford. I finally felt that I had time (partly through retirement) to dig deeper into scientific questions I had wondered about, and naturally explored topics associated with my own research. I therefore could have added the words “soft” to x-rays and “resonant” to interactions in the title of the book, but the present broader title is intended to pull in the entire x-ray community. I have tried to connect classical, quantum, and quantum electrodynamics (QED) concepts. Their unified formulation required repetition of conventional semi-classical treatments that may be found in other books, but what I consider novel is the introduction of QED and quantum optics concepts to x-ray science. Whatever area of x-ray science you come from, I hope that you will find the book pedagogical and interesting and sections of it novel and fascinating. I have found that some of the most difficult questions to answer are naive childlike questions since they often go to the core of things. The impetus for writing this book was partly my curiosity of understanding the nature of “light”, especially the concepts of coherence, interference, and diffraction, and how to describe x-ray interactions with matter. I completely underestimated the difficulty of doing so and repeatedly wondered how I could finish this book without understanding the very nature of “light” itself to my own satisfaction. I was consoled by Feynman’s words “Even the experts don’t understand it as well as they would like”. In physics, “understanding” typically means describing observations through mathematical formulations that through certain assumptions and symmetries reduce the infinite complexity of nature. In the process, I often wondered about alternative formulations, say by extraterrestrials. Writing this book was not a “brain dump” of my knowledge. In many cases, I first had to read the literature to learn the material, either from scratch or beyond a cursory understanding. Most of my time was spent on mathematical derivations of different phenomena—a lonesome endeavor—which resulted in long detailed drafts of chapters. Mind you, the joyous periods, when a solution emerged or a section had been finished, were interleaved with periods of agonizing over things. Even the search for a minus sign, a factor of 2, π or i can be frustrating, time consuming, and drive you crazy! On the other hand, I often thought about the privilege of being a scientist. Like artists, true scientists want to explore and create, but we are fortunate to get paid to pursue our passion and do not fall into the category of “starving artists”.

Preface

ix

I very much like how Steve Jobs described the creative process at a Stanford commencement speech: “When you first start off trying to solve a problem, the first solutions you come up with are very complex, and most people stop there. But if you keep going, and live with the problem and peel more layers of the onion off, you can often times arrive at some very elegant and simple solutions.” I often found myself working out a problem with full mathematical rigor, only to isolate its physical essence after stripping away or simplifying much of the math. I have strived to interpret the abstract mathematical formulations by simple mental pictures. It is these pictures that remain even after you have forgotten the details. I seriously started to work on this book after stepping down from management in 2013. Getting older, I wanted to pursue my passions rather than to deal with managerial responsibilities, which increasingly appeared to be “golden handcuffs”. My favorite days as a manager were those when my calendar allowed me to devote time to learning and writing and especially the “group meetings” with my students and collaborators. After “retiring”, I spent the winters at beautiful Red Mountain Resort in British Columbia, Canada, where periods of writing were interleaved with my other passion, deep powder downhill skiing, and classical and skating cross-country skiing with my wife Linda. One of the most significant realizations for me in writing this book was that the understanding of light (or the world around us) resembles an infinite series of steps that appears bottomless. I therefore had to draw the line at a certain level of my own understanding. In the end, I also had to make choices of what to include in the final version of the book. My detailed write-ups and derivations amounted to about 1500 pages in the format of this book, and I had to omit topics and leave out details of derivations. I did not formulate homework problems for students, hoping that the book provides plenty of material to entice the reader’s scientific curiosity. I personally have learned more from digging into curiosity-inspired problems than homework assignments. The final material chosen for the book naturally reflects my personal taste and expertise developed through my 50-year journey from synchrotron radiation to XFEL developments and research. As a scientist, whose first goal was opening up the soft x-ray region, previously plagued by a gap from about 250–3000 eV due to carbon contaminated optics at the low end and Be window transmission at the high end, I have admired and benefitted from the pioneering contributions of Burton Henke, largely based on laboratory-based x-ray sources, which culminated in the modern compilation of x-ray scattering factors and cross sections. They are now widely available in updated form on the web1 through the efforts of Eric Gullikson. My learning process was also aided by key books: D. Attwood’s and A. Sakdinawat’s X-Rays and Extreme Ultraviolet Radiation: Principles and Applications, J. Als-Nielsen and D. McMorrow’s Elements of Modern X-Ray Physics, J. J. Sakurai’s Modern Quantum Mechanics, J. W. Goodman’s books Fourier Optics and Statistical Optics, E. Hecht’s Optics, R. Loudon’s The Quantum Theory of Light and M. O.

1

https://henke.lbl.gov/optical_constants/.

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Scully and M. S. Zubairy’s Quantum Optics. I also spent considerable time learning quantum optics from other books and papers, referenced in the text. Of course, I learned from many people in person like my students, of whom Bill Schlotter and Diling Zhu became true x-ray “jocks”. I especially befitted from my more recent postdocs and collaborators, in particular, Ives Acremann, Hermann Dürr, Stefan Eisebitt, Jan Lüning, Hendrik Ohldag, Alexander Reid, and Andreas Scherz. Andreas sent me in a new direction when he showed me data that he and my student Benny Wu had measured at LCLS in 2012. Rather than turning the problem over to theoreticians, as is often done by experimentalists like me, I wanted to understand the experimental results on my terms. So I followed Andreas’ and my intuition about their meaning. This required learning quantum optics and formulating the phenomenon of diffraction in a photon picture. During this arduous journey, filled with periods of inspiration and self-doubts, I realized the existing gap between the conventional concepts used in the x-ray community and the modern QED formalism and language of quantum optics. I hope that the present book contributes to overcoming this barrier, especially because the two communities increasingly mingle in XFEL-based science. Most of all, I hope that the book inspires young scientists to push beyond my understanding. Over the years, I benefitted from discussions with (in alphabetical order) Massimo Altarelli, John Arthur, David Attwood, Uwe Bergmann, Martin Beye, Henry Chapman, Ryan Coffee, Georgi Dakovski, Hubert Ebert, Alexander Föhlisch, Kelly Gaffney, Frank De Groot, Giacomo Ghiringhelli, Janos Hajdu, Jerry Hastings, Olav Hellwig, Zhirong Huang, Ludvig Kjellsson, Jianbin Liu, Bob Nagler, Josef Nordgren, Nils Mårtensson, Martin Magnuson, Shaul Mukamel, Z. Y. (Jeff) Ou, Bruce Patterson, Lars Pettersson, Claudio Pellegrini, John Rehr, Andrei Rogalev, Jan-Erik Rubensson, Evgeny Saldin, Robin Santra, Bob Schoenlein, Sebastian Stepanow, Gerrit van der Laan, Michel Van Veenendaal, Ivan Vartanyants, and Linda Young. Special thanks to Frithjof Nolting and Pietro Gambardella for sponsoring my sabbatical at PSI/ETH and Pietro for allowing me to use his beautiful experimental data, to Faris Gel’mukhanov for sharing his insights and detailed derivations of RIXS phenomena, to Joseph Goodman who patiently helped me with various optics problems over the years, and finally to my good friend Anders Nilsson for many exchanges on science, philosophy, and spirituality. Possible corrections and updates of the book will be posted on my website [3]. Emerald Hills, CA, USA

Joachim Stöhr

References 1. J. Stöhr, NEXAFS Spectroscopy (Springer, Heidelberg, 1992) 2. J. Stöhr, H.C. Siegmann, Magnetism: From Fundamentals to Nanoscale Dynamics (Springer, Heidelberg, 2006) 3. https://stohr.sites.stanford.edu/

Contents

1

Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 About the Present Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Nature of Light: From Light Rays to QED . . . . . . . . . . . . . . 1.2.1 Early Concepts of the Nature of Light . . . . . . . . . . . . . 1.2.2 The Quantum Era and Wave-Particle Duality . . . . . . . 1.2.3 New Insight in the Period 1930–1950 . . . . . . . . . . . . . . 1.2.4 Beyond Quantum Mechanics: Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Development of the Laser and Quantum Optics Around 1960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The X-Ray Revolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 The Discovery and Early Utilization of X-Rays . . . . . 1.3.2 Development of Synchrotron Radiation Sources . . . . . 1.3.3 The Advent of X-Ray Free Electron Lasers . . . . . . . . . 1.4 The Scientific Power of X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 From Optical to X-Ray Response of Matter . . . . . . . . 1.4.2 The Importance of X-Ray Resonances . . . . . . . . . . . . . 1.4.3 X-Ray Spectro-Microscopy . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Summary of Key Capabilities of Synchrotron Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Science with XFELs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Snapshots of the Atomic Structure of Matter: Probe Before Destruction . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2 Creation and Characterization of Transient States of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 6 6 8 10 11 12 14 14 17 19 24 24 27 32 35 36 36 38

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1.5.3

Creation and Probing High Energy Density Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Non-linear X-Ray Interactions with Matter . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I 2

42 44 50

Production of X-Rays and Their Description

Production of X-Rays: From Virtual to Real Photons . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Relativistic Concepts and Electron Bunch Compression . . . . . . . 2.3 The Fields of a Moving Charge: Liénard–Wiechert Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Fields of a Charge in Uniform Motion: Velocity Fields . . . . . . . . 2.4.1 Spatial Dependence of Velocity Fields of a Single Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Temporal Dependence of Velocity Fields of a Single Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 The Fields of a Relativistic Gaussian Electron Bunch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Generation of Huge Field Pulses: THz Fields and Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Frequency Spectrum of Gaussian Electron Bunches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Weizsäcker-Williams Method: Virtual Photon Spectrum . . . . . . 2.5.1 Virtual Photon Spectrum of a Gaussian Electron Bunch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Coherent Virtual Spectrum: THz Photons . . . . . . . . . . 2.5.3 Incoherent Virtual Spectrum: X-Ray Photons . . . . . . . 2.6 Acceleration Fields: Creation of EM Radiation . . . . . . . . . . . . . . 2.6.1 Distortion of Field Lines: Radiation . . . . . . . . . . . . . . . 2.6.2 The Angular Spectrum of a Single Accelerated Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Frequency Spectrum of a Single-Electron Bending Magnet Source . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 Frequency Spectrum of a Single-Electron Undulator Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 The Angular Distribution of Undulator Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Synchrotron Radiation Spectrum of Electron Bunches . . . . 2.7.1 The Spectrum of an Electron Bunch and Its Temporal Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 The Emitted Radiation Cone and Lateral Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Toward Ultimate Storage Rings . . . . . . . . . . . . . . . . . . . 2.7.4 Polarization of Synchrotron Radiation . . . . . . . . . . . . .

61 61 63 65 68 68 71 72 76 77 79 80 81 84 85 87 88 92 96 99 101 101 101 104 105

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2.8

X-Ray Free Electron Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 The Radiation Emitted by SASE XFELs . . . . . . . . . . . 2.8.2 Realistic Pulse Structure in SASE XFELs . . . . . . . . . . 2.8.3 From SASE to Transform Limited Pulses . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

108 109 112 114 115

From Electromagnetic Waves to Photons . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Classical Description of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Maxwell’s Equations and Their Symmetries . . . . . . . . 3.2.2 The Electromagnetic Wave Equation . . . . . . . . . . . . . . 3.2.3 Energy, Momentum, Intensity, Flux, and Fluence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Field Strength of X-Ray Beams . . . . . . . . . . . . . . . . . . . 3.2.5 Field Normalization Volume . . . . . . . . . . . . . . . . . . . . . 3.2.6 The Description of Polarized EM Waves . . . . . . . . . . . 3.2.7 The Degree of X-Ray Polarization . . . . . . . . . . . . . . . . 3.3 Quantum Theoretical Description of X-Rays . . . . . . . . . . . . . . . . 3.3.1 The Quantized Electromagnetic Field: Birth of the Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Zero-Point Energy and Virtual Photons . . . . . . . . . . . . 3.3.3 The Renormalized Hamiltonian of the Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4 The Basic Quantum States of Light . . . . . . . . . . . . . . . . 3.3.5 Single Mode Number States . . . . . . . . . . . . . . . . . . . . . . 3.4 The Properties of Single Mode Quantum States . . . . . . . . . . . . . . 3.4.1 Definition of Key Properties . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Properties of Single Mode Number States . . . . . . . . . . 3.4.3 Properties of Single Mode Coherent States . . . . . . . . . 3.4.4 Properties of Single Mode Chaotic States . . . . . . . . . . 3.5 Photon Modes, Density of States, and Coherence Volume . . . . . 3.5.1 Photon Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Number of Modes per Unit Energy . . . . . . . . . . . . . . . . 3.5.3 Number of Modes per Unit Volume . . . . . . . . . . . . . . . 3.5.4 Coherence Volume per Mode . . . . . . . . . . . . . . . . . . . . . 3.5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Link of Classical and Quantum Properties of Radiation . . . . . . . 3.6.1 Multi-mode Number States . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Expectation Value of the Squared Electric Field . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 120 120 121

Brightness and Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Brightness and Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Introduction to the Concept of Brightness . . . . . . . . . . 4.2.2 Formal Definition of Brightness . . . . . . . . . . . . . . . . . .

159 159 160 160 161

3

4

122 125 126 127 131 132 132 136 136 139 139 140 141 142 143 146 148 148 150 151 151 152 153 153 154 157

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4.2.3 4.2.4 4.2.5

5

Average Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peak Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brightness Reduction Through Partial Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6 Link of Brightness and Coherence . . . . . . . . . . . . . . . . 4.2.7 Photon Degeneracy Parameter . . . . . . . . . . . . . . . . . . . . 4.2.8 Brightness of Storage Rings and XFELs . . . . . . . . . . . 4.2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Historical Descriptions of Coherence . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Geometrical Coherence Concepts in Time and Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Fourier Optics Description of Coherent Sources . . . . . . . . . . . . . 4.4.1 Fourier Transformation of Temporal Coherence . . . . . 4.4.2 Time-Bandwidth Product Definitions . . . . . . . . . . . . . . 4.4.3 Fourier Transformation of Spatial Coherence . . . . . . . 4.5 Statistical and Quantum Description of Partially Coherent Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Degree of First Order Coherence . . . . . . . . . . . . . . . . . . 4.5.2 Statistical Forms of Light Behavior . . . . . . . . . . . . . . . . 4.5.3 The Concept of Partial Lateral Coherence . . . . . . . . . . 4.6 The van Cittert–Zernike Theorem: Propagation of Field Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 van Cittert–Zernike Theorem for a Schell Model Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Coherent Limit of a Circular Flat-Top Source . . . . . . . 4.6.3 Chaotic Limit of a Circular Flat-Top Source . . . . . . . . 4.6.4 Coherent and Chaotic Limits of a Circular Gaussian Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Quantum Derivation of the van Cittert–Zernike Theorem . . . . . . 4.7.1 The Case of Two Source Points . . . . . . . . . . . . . . . . . . . 4.7.2 Finite Area Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Comparison of Classical and Quantum Diffraction . . . . . . . . . . . 4.8.1 Link of Waves and One-Photon Probability Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 Link of Probability Amplitude and Operator Formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 X-Ray Measurement of First Order Lateral Coherence . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

164 166

The Complete Description of Light: Higher Order Coherence . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Existence of Photons and Their Interference . . . . . . . . . . . . . 5.2.1 The Grangier–Roger–Aspect Experiment . . . . . . . . . . 5.2.2 The Hanbury Brown–Twiss Experiment . . . . . . . . . . . . 5.2.3 The Hong–Ou–Mandel Experiment . . . . . . . . . . . . . . .

227 227 229 229 230 232

167 170 173 173 175 176 177 180 180 184 186 192 195 196 198 199 202 203 205 208 210 210 211 217 219 219 220 221 224

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235 235 237 238 238 239 240 241

Light and Detection Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 “Light” Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Detector Timescales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Description of Second Order Coherence . . . . . . . . . . . . . . . . 5.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Second Order Spatial Coherence Functions . . . . . . . . . 5.4.3 Link of First and Second Order Coherence . . . . . . . . . 5.5 The Propagation of Second Order Coherence . . . . . . . . . . . . . . . . 5.5.1 Propagation of Two Photons from Two Source to Two Detection Points . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Fundamental Two-Photon Patterns of Two-Point Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 The New Paradigm of Quantum Diffraction . . . . . . . . 5.5.4 Finite Source Areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.5 Second Order Pattern of a Circular Flat-Top Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.6 Power Conservation in Two-Photon Diffraction . . . . . 5.5.7 X-Ray Measurements of Second Order Lateral Coherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Detection of First and Second Order Coherence Patterns . . . . . . 5.6.1 The Coherent Quantum States of Light . . . . . . . . . . . . 5.6.2 Implication for the First Order Pattern . . . . . . . . . . . . . 5.6.3 Implications for the Second Order Pattern . . . . . . . . . . 5.7 Higher Order Coherence Propagation . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Higher Order Patterns of a Coherent Source . . . . . . . . 5.7.2 Higher Order Patterns of a Chaotic Flat-Top Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Higher Order Brightness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 From Rays to Waves to Photons to Rays—Going Full Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241 244 250 250 251 253 256 259 259 262 263 266 267 270 272 275 277

Part II Semi-classical Theory of X-Ray Interactions with Matter 6

Semi-classical Response of Atoms to Electromagnetic Fields . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 X-Ray Thomson Scattering by Electrons and Spins . . . . . . . . . . 6.2.1 Response of an Electron and Its Spin to the Incident Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 The Scattered Dipole Field . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Thomson Scattering Length and Cross Sections . . . . . 6.2.4 Thomson Scattering By An Atom . . . . . . . . . . . . . . . . .

283 283 284 284 286 288 291

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6.3

X-Ray Resonant Scattering and Absorption . . . . . . . . . . . . . . . . . 6.3.1 Dispersion Corrections to the Atomic Scattering Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Resonant Scattering by Atomic Core Electrons . . . . . . 6.3.3 Distinction of Thomson, Rayleigh, and Resonant Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Classical Link of Resonant Scattering and Absorption . . . . 6.4.1 Relative Size of Thomson, Resonant Scattering, and Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The First Born Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 The Breit–Wigner Atomic Cross Section . . . . . . . . . . . 6.6 The Kramers-Kronig Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 The Henke-Gullikson Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Atomic Shell Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Semi-classical Response of Solids to Electromagnetic Fields . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Static Magnetic and Electric Fields Inside Materials . . . . . . . . . . 7.3 Frequency Response of Materials: Microwaves to X-Rays . . . . . 7.3.1 Permittivity: Electric Field Response . . . . . . . . . . . . . . 7.3.2 Permeability: Magnetic Field Response . . . . . . . . . . . . 7.3.3 From Permittivity and Permeability to Optical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.4 Penetration Depth of EM Waves: From Microwaves to X-Rays . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.5 Optical Parameters: From the THz to the X-Ray Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Dielectric Response Formulation of X-Ray Absorption . . . . . . . 7.4.1 Optical Parameter Formulation . . . . . . . . . . . . . . . . . . . 7.4.2 Absorption Coefficient Formulation . . . . . . . . . . . . . . . 7.4.3 Beer-Lambert Formulation . . . . . . . . . . . . . . . . . . . . . . . 7.5 From Dielectric to Atom-Based X-Ray Response . . . . . . . . . . . . 7.5.1 Brief Review of X-Ray Scattering Factors . . . . . . . . . . 7.5.2 From Single Atoms to Atomic Sheets . . . . . . . . . . . . . . 7.5.3 Atomic Scattering Factors: Born Approximation Versus Huygens–Fresnel Principle . . . . . . . . . . . . . . . . 7.5.4 Scattering Phase Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.5 Link of X-Ray Scattering Factors and Optical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 The Optical Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Coherent Versus Incoherent X-Ray Scattering from Solids . . . . 7.7.1 Incoherent Versus Coherent Field Superposition . . . . . 7.7.2 X-Ray Forward Scattering and Absorption . . . . . . . . . 7.7.3 The Total Transmitted Intensity . . . . . . . . . . . . . . . . . . . 7.7.4 Relative Size of the Absorbed and Scattered Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

293 294 296 298 299 301 302 304 305 309 313 315 317 317 318 320 320 321 323 325 327 330 330 332 333 334 334 335 337 339 341 343 344 345 347 349 350

Contents

The Response of a Thick Sample: Dynamical Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.8.1 Snell’s Law and Total X-Ray Reflection . . . . . . . . . . . . 7.8.2 Darwin-Prins Dynamical Scattering Theory . . . . . . . . 7.8.3 The Transmitted Field and Intensity . . . . . . . . . . . . . . . 7.8.4 The Reflected Field in the Soft X-Ray Region . . . . . . 7.9 Polarization Dependent Absorption: Dichroism . . . . . . . . . . . . . . 7.9.1 History of Polarization Dependent Effects . . . . . . . . . . 7.9.2 Chiral Versus Magnetic Orientation . . . . . . . . . . . . . . . 7.9.3 Fundamental Forms of X-Ray Dichroism . . . . . . . . . . 7.10 Natural Dichroism and Orientational Order . . . . . . . . . . . . . . . . . 7.10.1 Orientation Factors and Saupe Matrix . . . . . . . . . . . . . 7.10.2 Determination of Orientation Factors . . . . . . . . . . . . . . 7.11 Magnetic Dichroism and Faraday Rotation . . . . . . . . . . . . . . . . . . 7.11.1 Polarization Dependent Scattering Length and Optical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11.2 Phenomenological Model . . . . . . . . . . . . . . . . . . . . . . . . 7.11.3 Transmission of Circularly Polarized X-Rays: XMCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11.4 Transmission of Linearly Polarized X-Rays: XMLD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.11.5 Faraday Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8

Classical Diffraction and Diffractive Imaging . . . . . . . . . . . . . . . . . . . . 8.1 Introduction and Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Real Space X-Ray Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 X-Ray Microscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Polarization Dependent Microscopy . . . . . . . . . . . . . . . 8.3 Diffractive Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Introduction to Diffractive Imaging . . . . . . . . . . . . . . . . 8.3.2 Historical Development of Diffraction Theories . . . . . 8.3.3 The Huygens–Fresnel Principle . . . . . . . . . . . . . . . . . . . 8.3.4 The Rayleigh-Sommerfeld Diffraction Formula . . . . . 8.4 Approximate Solutions of the Rayleigh-Sommerfeld Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Paraxial Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Fresnel Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Fraunhofer Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Fourier Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Diffraction and Wave-Particle Duality . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Feynman’s Probability Amplitudes . . . . . . . . . . . . . . . . 8.5.2 De Broglie–Bohm Pilot Wave Theory . . . . . . . . . . . . .

351 352 353 356 358 359 359 360 362 364 366 369 371 372 373 375 378 379 382 385 385 387 388 391 392 392 395 397 400 402 402 403 404 404 405 406 408

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8.6

Consequences of Fraunhofer Diffraction . . . . . . . . . . . . . . . . . . . . 8.6.1 The Diffraction Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 The Arago-Fresnel-Poisson Bright Spot . . . . . . . . . . . . 8.6.3 Babinet’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Formulation of Polarization Dependent Diffractive Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Charge Domains with Orientational Order . . . . . . . . . . 8.7.2 Charge Domains of Different Chemical Composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.3 Diffraction by Ferromagnetic Domains . . . . . . . . . . . . 8.7.4 Separation of Charge and Magnetic Contrast . . . . . . . . 8.8 Illustration of the Phase Problem in X-Ray Diffractive Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9 Fourier Transform Holography: FTH . . . . . . . . . . . . . . . . . . . . . . . 8.9.1 Illustration of FTH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.9.2 Improved Reference Beams . . . . . . . . . . . . . . . . . . . . . . 8.9.3 Application of FTH: Magnetic Domains . . . . . . . . . . . 8.10 Non-holographic Solutions to the Phase Problem . . . . . . . . . . . . 8.10.1 Brief History of X-Ray Crystallography . . . . . . . . . . . . 8.10.2 MIR, SAD, and MAD Image Reconstruction . . . . . . . 8.10.3 Sampling of the Diffraction Pattern . . . . . . . . . . . . . . . . 8.10.4 Ptychography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11 Multiple-Wavelength Anomalous Diffraction—MAD . . . . . . . . . 8.11.1 MAD of Macromolecular Crystals . . . . . . . . . . . . . . . . 8.11.2 Formulation of MAD in Protein Cystallography . . . . . 8.11.3 MAD Imaging of Non-crystalline Samples . . . . . . . . . 8.11.4 Implementation of MAD for Non-periodic Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11.5 Phase Contrast Imaging: Combining FTH and MAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

408 409 412 414 415 417 421 422 426 428 431 432 436 438 441 442 443 444 446 447 448 448 452 454 459 460

Part III Quantum Theory of Weak Interactions 9

Quantum Formulation of X-Ray Interactions with Matter . . . . . . . . 9.1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Photon-Matter Interaction Hamiltonian . . . . . . . . . . . . . . . . . 9.2.1 The Pauli Equation Including the EM Field . . . . . . . . . 9.2.2 Evaluation of the Spin Dependent Part of the Pauli Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 The Complete Interaction Hamiltonian . . . . . . . . . . . . . 9.2.4 Relative Size of the Interactions . . . . . . . . . . . . . . . . . .

467 467 468 468 469 471 472

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Perturbation Treatment of X-Ray Scattering and Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 On the Use of Time-Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Kramers-Heisenberg-Dirac Perturbation Theory . . . . . . . . . . . . . 9.4.1 The Kramers-Heisenberg-Dirac Formula . . . . . . . . . . . 9.5 Overview of First Order Processes . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 X-Ray Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 Resonant X-Ray Absorption . . . . . . . . . . . . . . . . . . . . . 9.5.3 X-Ray Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.4 X-Ray Thomson Scattering . . . . . . . . . . . . . . . . . . . . . . 9.6 Overview of Second Order Processes . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Spontaneous X-Ray Resonant Scattering . . . . . . . . . . . 9.6.2 Stimulated X-Ray Resonant Scattering . . . . . . . . . . . . . 9.6.3 Two-Photon Absorption and Photoemission . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10 Quantum Theory of X-Ray Absorption Spectroscopy . . . . . . . . . . . . . 10.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Quantum Formulation of X-Ray Absorption Spectroscopy (XAS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1 Photon Flux, Intensity, and Absorption Cross Section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Non-resonant Absorption: Excitation into Continuum States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1 Wavefunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Continuum Cross Section . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 Simple Model Calculation . . . . . . . . . . . . . . . . . . . . . . . 10.3.4 The Core Level Photoemission Spectrum and Its Linewidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Resonant X-Ray Absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Natural Linewidth of XAS Resonances . . . . . . . . . . . . 10.4.2 The Natural Shape of XAS Resonances . . . . . . . . . . . . 10.4.3 Natural Linewidth of Optical Versus X-Ray Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.4 The Dipole Matrix Element . . . . . . . . . . . . . . . . . . . . . . 10.4.5 One-Electron/Hole Model . . . . . . . . . . . . . . . . . . . . . . . 10.4.6 Polarization Dependence of the Angular Transition Matrix Element . . . . . . . . . . . . . . . . . . . . . . . 10.4.7 Sum Rules for the Angular Transition Matrix Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Resonant XAS in Experiment and Theory . . . . . . . . . . . . . . . . . . 10.5.1 K-Shell Resonance in the Low-Z Atom Ne . . . . . . . . . 10.5.2 K-Shell Resonances in the N2 and O2 Molecules . . . . 10.5.3 L-Shell Resonances in 3d Transition Metal Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

474 474 477 478 480 480 481 482 483 484 484 485 486 486 489 489 489 491 492 492 494 496 497 499 499 501 503 504 505 508 509 513 513 515 517

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10.5.4

L-Shell XAS Intensities and Valence Shell Occupation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.5 Resonant Lineshapes in Atoms and Solids . . . . . . . . . . 10.5.6 Dipole Matrix Element, Oscillator Strength, and Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 Multi-electron Formalism: Multiplet Structure . . . . . . . . . . . . . . . 10.6.1 Evolution of One-Electron to Multiplet Theory . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Quantum Theory of X-Ray Dichroism . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Introduction to the Quantum Theory of Dichroism . . . . . . . . . . . 11.3 X-Ray Natural Linear Dichroism—XNLD . . . . . . . . . . . . . . . . . . 11.3.1 The Search Light Effect . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 XNLD and the Quadrupolar Valence Charge Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.3 Application of XNLD . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 X-Ray Natural Circular Dichroism—XNCD . . . . . . . . . . . . . . . . 11.4.1 The Two Types of XNCD . . . . . . . . . . . . . . . . . . . . . . . . 11.5 X-Ray Magnetic Circular Dichroism—XMCD . . . . . . . . . . . . . . 11.5.1 Key Concepts of Magnetism and Magnetic Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 XMCD Sum Rule for the Orbital Moment . . . . . . . . . . 11.5.3 Experimental Studies of Orbital Magnetism . . . . . . . . 11.5.4 XMCD Sum Rule for the Spin Moment . . . . . . . . . . . . 11.6 Test of the Sum Rules: Cu-Phthalocyanine . . . . . . . . . . . . . . . . . . 11.6.1 Electronic Structure of Cu-Pc . . . . . . . . . . . . . . . . . . . . 11.6.2 Treatment of the Spin-Orbit Interaction . . . . . . . . . . . . 11.6.3 Comparison of Orbital Momenta in Theory and Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.6.4 Spin Momenta in Theory and Experiment . . . . . . . . . . 11.7 Application of XMCD to the Study of Transient Spin Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.1 Spin Accumulation in Cu upon Injection from Co . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.7.2 Spin-Orbit Induced Spin Currents in Pt, Injected into Co . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8 X-Ray Magnetic Linear Dichroism—XMLD . . . . . . . . . . . . . . . . 11.8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8.2 Theoretical Formulation in One-Electron Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.8.3 XNLD Versus XMLD in Cu-Phthalocyanine . . . . . . . . 11.8.4 XMLD in Ferromagnetic Transition Metals . . . . . . . . . 11.8.5 Enhanced XMLD Through Multiplet Effects . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

522 523 525 527 529 534 537 537 537 540 541 543 546 547 550 552 552 559 562 564 566 566 568 573 574 577 578 581 582 582 584 585 586 588 590

Contents

12 Quantum Theory of X-Ray Emission and Thomson Scattering . . . . 12.1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Quantum Formulation of X-Ray Emission Spectroscopy (XES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 XES History and Terminology . . . . . . . . . . . . . . . . . . . . 12.2.2 The Photon Part of the Transition Matrix Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.3 XES Decay Time and Linewidth . . . . . . . . . . . . . . . . . . 12.2.4 Decays to Excited Final States . . . . . . . . . . . . . . . . . . . . 12.2.5 Auger Contribution to the XES Linewidth . . . . . . . . . . 12.2.6 Putting It All Together: The X-Ray Emission Rate and Width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.7 Atomic Decay Time: Core Hole-Clock . . . . . . . . . . . . . 12.3 Fundamental X-Ray Emission Experiments . . . . . . . . . . . . . . . . . 12.3.1 K-shell Emission in Ne . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.2 K-Shell Emission in N2 . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.3 L-Shell XES in 3d Metals . . . . . . . . . . . . . . . . . . . . . . . 12.3.4 L3 -Shell XES in Cu Metal . . . . . . . . . . . . . . . . . . . . . . . 12.4 X-Ray Fluorescence Yield, Linewidths, and Strengths . . . . . . . . 12.4.1 Radial Dipole Matrix Element . . . . . . . . . . . . . . . . . . . . 12.5 Quantum Theory of Thomson Scattering . . . . . . . . . . . . . . . . . . . 12.5.1 Quantum Theoretical Formulation of Thomson Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Elastic Thomson Scattering: Atomic Form Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.3 Inelastic Thomson Scattering: Dynamical Structure Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.4 Core Shell Excitations: X-Ray Raman Scattering (XRS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.5.5 Example: Typical O K-shell Cross Sections . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Quantum Theory of X-Ray Resonant Scattering . . . . . . . . . . . . . . . . . 13.1 Introduction and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Formulation of Resonant Scattering: REXS and RIXS . . . . . . . . 13.2.1 Evaluation of the Double Matrix Element . . . . . . . . . . 13.2.2 One-Electron Versus Configuration Picture . . . . . . . . . 13.2.3 Coherent Second Order Versus Consecutive First Order Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2.4 REXS/RIXS Terminologies . . . . . . . . . . . . . . . . . . . . . . 13.3 Quantum Formulation of REXS . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3.1 The Fundamental REXS Cross Section . . . . . . . . . . . . 13.3.2 REXS with Finite Instrumental Resolution . . . . . . . . . 13.4 Spontaneous and Stimulated REXS Versus XAS . . . . . . . . . . . . . 13.4.1 Spontaneous REXS Versus XAS . . . . . . . . . . . . . . . . . .

xxi

595 595 596 596 597 599 601 602 604 605 607 607 609 611 611 614 614 616 618 621 622 623 628 628 631 631 632 633 636 637 639 641 641 642 643 644

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Contents

13.4.2 Stimulated REXS Versus XAS . . . . . . . . . . . . . . . . . . . 13.4.3 Link to Semi-classical Results . . . . . . . . . . . . . . . . . . . . 13.5 Intermediate State Interference Effects in REXS . . . . . . . . . . . . . 13.5.1 REXS Interference Contour Map . . . . . . . . . . . . . . . . . 13.5.2 REXS Scattering Time . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.3 REXS Interference in Molecular Spectra: N2 and O2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.6 Polarization and Spin Dependent Spontaneous REXS . . . . . . . . 13.6.1 The Polarization Dependent Scattering Length . . . . . . 13.7 Spontaneous Versus Stimulated REXS by an Atomic Sheet . . . . 13.7.1 Forward Scattering by an Atomic Sheet . . . . . . . . . . . . 13.8 Resonant Inelastic X-Ray Scattering: RIXS . . . . . . . . . . . . . . . . . 13.8.1 Two-Step RIXS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.9 RIXS with Finite Instrumental Resolution . . . . . . . . . . . . . . . . . . 13.9.1 The Case of Small Final State Width . . . . . . . . . . . . . . 13.9.2 Reduction of RIXS to XES . . . . . . . . . . . . . . . . . . . . . . . 13.10 Examples of RIXS Capabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.10.1 K-Shell RIXS of N2 and O2 . . . . . . . . . . . . . . . . . . . . . . 13.10.2 L-edge RIXS of Transition Metal Oxides . . . . . . . . . . . 13.10.3 L-Edge RIXS of Transition Metals . . . . . . . . . . . . . . . . 13.10.4 RIXS of Chemisorbed Molecules: Polarization Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.10.5 Utilization of the Scattered Polarization . . . . . . . . . . . . 13.11 RIXS and Reduced Linewidth XAS (HERFD) . . . . . . . . . . . . . . . 13.11.1 HERFD XAS at the Pt L3 -Edge . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

647 647 648 651 651 653 656 657 661 662 663 664 666 667 668 668 668 673 676 681 684 685 686 689

Part IV Multi-photon Interaction Processes 14 Resonant Non-linear X-Ray Processes in Atoms . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 X-Ray Induced Atomic Core to Valence Transitions . . . . . . . . . . 14.2.1 Interaction Energy and Hamiltonian: The Rabi Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 The Rabi Frequency in the X-Ray Regime . . . . . . . . . . 14.3 The Optical Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.1 Time-Dependent Transitions in a Two-Level System: Density Matrix Formulation . . . . . . . . . . . . . . 14.3.2 Damping Constants: Longitudinal Versus Transverse Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Definition of Transition Rates in the BR Theory . . . . . . . . . . . . . 14.4.1 X-Ray Interaction Parameters for Model Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.2 Practical Units and Beam Parameter Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

695 695 697 698 699 701 702 704 705 706 707

Contents

14.5

Analytical Solutions of the Bloch Equations . . . . . . . . . . . . . . . . . 14.5.1 Arbitrary Bandwidth: Low Incident Intensity . . . . . . . 14.5.2 Exact Resonance: Arbitrary Incident Intensity . . . . . . 14.5.3 Excitations by Transform-Limited and SASE Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.4 Solution for the Steady-State or Long Time Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.5 Power Broadening of the BR Linewidth . . . . . . . . . . . . 14.6 Link of KHD and Low Intensity BR Rates . . . . . . . . . . . . . . . . . . 14.6.1 The KHD Transition Rates . . . . . . . . . . . . . . . . . . . . . . . 14.6.2 Link of Bandwidth in KHD and Time in BR Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.3 Mode-Based Versus Atom-Based Coherence Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.4 Zero-Point Field in the Bloch-Rabi Formalism . . . . . . 14.7 Link of BR Rates and KHD Rates in the Steady-State . . . . . . . . 14.7.1 Steady-State Rate Expressions . . . . . . . . . . . . . . . . . . . . 14.7.2 Illustration of the BR Rates and Their Saturation . . . . 14.7.3 Time Dependence of Rates at Resonance . . . . . . . . . . . 14.8 Optical Theorem: Sum Rule for Absorption and Scattering . . . . 14.8.1 XAS and REXS Cross-Section Sum Rule . . . . . . . . . . 14.8.2 Atom Transmission Sum Rule . . . . . . . . . . . . . . . . . . . . 14.8.3 BR Versus KHD Stimulated Enhancement: Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9 BR, KHD, and Einstein Treatment of a Two-Level System . . . . 14.9.1 Einstein’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.9.2 Reduction of the BR to the Einstein Theory . . . . . . . . 14.10 Resonance Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.10.2 Second Quantization of the p · A Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.10.3 The Degree of First Order Temporal Coherence . . . . . 14.10.4 g (1) (t) From Numerical Solutions of Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.11 The Resonant Fluorescence Spectrum . . . . . . . . . . . . . . . . . . . . . . 14.11.1 Form of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.11.2 Fourier Transform of g (1) (t): The Fluorescence Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.11.3 The Low Intensity Spectrum: R < /4 . . . . . . . . . . . 14.11.4 High Intensity Spectrum: R > /4 . . . . . . . . . . . . . . . 14.11.5 Calculated Fluorescence Spectra . . . . . . . . . . . . . . . . . . 14.11.6 Coherent and Incoherent Parts of the Equilibrium Spectrum . . . . . . . . . . . . . . . . . . . . . 14.12 Second Order Coherence of the Fluorescent Photons . . . . . . . . . 14.12.1 Weak Incident Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxiii

708 709 711 714 715 718 719 720 721 722 725 726 726 728 731 732 733 734 735 736 736 739 742 742 744 745 747 749 749 750 751 752 752 753 756 757

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Contents

14.12.2 Large Dephasing: Einstein Result . . . . . . . . . . . . . . . . . 14.12.3 Resonant Case of Arbitrary Intensity . . . . . . . . . . . . . . 14.12.4 Photon Antibunching in Resonance Fluorescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

758 759

15 Non-linear Absorption and Scattering Processes in Solids . . . . . . . . . 15.1 Introduction and Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.2 Chapter Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2 The Fundamental Damage Issue of XFEL Radiation . . . . . . . . . . 15.2.1 X-Ray Beam Parameters . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Temporal Evolution of Matter after X-Ray Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.3 Energy Transfer to the Electronic System . . . . . . . . . . 15.2.4 From Electronic to Lattice Temperature . . . . . . . . . . . . 15.2.5 Ablation Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3 Fluence-Dependent Changes of XAS Spectra . . . . . . . . . . . . . . . . 15.3.1 Redistribution of Valence Electrons . . . . . . . . . . . . . . . 15.3.2 X-Ray Transparency: An Introduction . . . . . . . . . . . . . 15.4 BR Theory of the Stimulated Response of a Thin Sheet . . . . . . . 15.4.1 Non-linear Response of an Atomic Sheet . . . . . . . . . . . 15.4.2 Effective Excited State Population and Enhancement Factor Gcoh . . . . . . . . . . . . . . . . . . . . 15.5 Non-linear Transmission Through a Film of Finite Thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 From Thin Sheet to Finite Thickness Film . . . . . . . . . . 15.5.2 Summary: From Single Atom to Film Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.3 Sum Rule for Non-linear Film Transmission . . . . . . . . 15.6 Polarization and Time-Dependent NL Transmission . . . . . . . . . . 15.6.1 The Polarization Dependent Generalized Beer-Lambert Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6.2 Non-linear Polarization Dependent Transmission by the Magnetic 3d Metals . . . . . . . . . . . 15.6.3 Dependence on X-Ray Pulse Coherence Time . . . . . . 15.6.4 From Collective to Independent Atomic Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7 X-Ray Transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.1 Resonant Case: Co Metal . . . . . . . . . . . . . . . . . . . . . . . . 15.7.2 Resonant Versus Non-resonant X-Ray Transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.3 Non-resonant Transparency Above the Al L-Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.7.4 Non-linear Transparency Above the Fe K-Edge . . . . .

765 765 765 765 767 768

761 762

768 770 774 775 777 777 777 780 782 783 784 785 787 792 793 796 796 798 800 802 803 804 807 809 811

Contents

15.8

Polarization Dependent NL Transmission at Resonance . . . . . . . 15.8.1 The Maximum NL Transmission Effect . . . . . . . . . . . . 15.8.2 Polarization Dependent Transmission . . . . . . . . . . . . . . 15.9 Competition Between NL Transmission and Diffraction . . . . . . 15.9.1 The NL Airy Pattern of a Film in a Circular Aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9.2 Change of the Spontaneous to the Stimulated Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.9.3 X-Ray Soliton Model: Mode-Dependent Stimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.10 Polarization Dependent NL Diffraction . . . . . . . . . . . . . . . . . . . . . 15.10.1 NL Diffraction by Magnetic Domains . . . . . . . . . . . . . 15.10.2 Narrow Bandwidth Resonant Case . . . . . . . . . . . . . . . . 15.10.3 Broad Bandwidth Case . . . . . . . . . . . . . . . . . . . . . . . . . . 15.11 Stimulated Resonant Inelastic X-Ray Scattering . . . . . . . . . . . . . 15.11.1 Stimulated L3 REXS and RIXS for Co Metal . . . . . . . 15.11.2 Observation of Stimulated RIXS in a Solid . . . . . . . . . 15.11.3 The Stimulated REXS/RIXS Model . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Quantum Diffraction: Emergence of the Quantum Substructure of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Generation of Different States of Light . . . . . . . . . . . . . . . . . . . . . 16.3 The Formulation of Quantum Diffraction . . . . . . . . . . . . . . . . . . . 16.3.1 First Order Diffraction Formulation . . . . . . . . . . . . . . . 16.3.2 Second Order Diffraction Formulation . . . . . . . . . . . . . 16.3.3 Order-Dependent Degree of Coherence . . . . . . . . . . . . 16.4 The Quantum States of Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.1 Two-Mode Collective Quantum States . . . . . . . . . . . . . 16.4.2 The Collective Coherent State and Its Substates . . . . . 16.4.3 The Collective Phase-Diffused Coherent State and Its Substates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.4.4 The Collective Chaotic State and Its Substates . . . . . . 16.4.5 Plots of the Substate Distributions . . . . . . . . . . . . . . . . . 16.4.6 Other Fundamental Quantum States . . . . . . . . . . . . . . . 16.4.7 Summary of Key Multi-photon Quantum States . . . . . 16.5 First Order Double-Slit Diffraction Patterns . . . . . . . . . . . . . . . . . 16.5.1 Calculation of the First Order Patterns . . . . . . . . . . . . . 16.5.2 Coherent State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.3 Plots of the First Order Patterns . . . . . . . . . . . . . . . . . . . 16.5.4 Degree of First Order Coherence . . . . . . . . . . . . . . . . . . 16.5.5 Reduction of First-Order Quantum to Wave Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxv

813 813 814 815 817 820 822 825 826 830 832 834 834 836 840 845 849 849 851 853 855 855 856 856 857 858 858 859 860 861 861 862 862 863 866 868 868

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Contents

16.6

Second Order Double-Slit Diffraction Patterns . . . . . . . . . . . . . . . 16.6.1 Coherent State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6.2 Plots of the Second Order Patterns . . . . . . . . . . . . . . . . 16.6.3 Degree of Second Order Coherence . . . . . . . . . . . . . . . 16.6.4 The Evolution from First to Second Order . . . . . . . . . . 16.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

869 870 871 874 875 876 877

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 881 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 911

About the Author

Joachim Stöhr received his Ph.D. from TU Munich, Germany, and, after spending time at Exxon and IBM Research Labs, joined Stanford University as a professor of Photon Science in 2000. He was the director of the Stanford Synchrotron Radiation Lightsource (2005–09) and the founding director of the Linac Coherent Light Source (2009–13). He has written two prior books, NEXAFS Spectroscopy (Springer, 1992) and Magnetism: From Fundamentals to Nanoscale Dynamics (Springer, 2006) with H. C. Siegmann. In 2011, he received the Davisson-Germer Prize in Surface Physics from the American Physical Society. He has been a professor emeritus of Photon Science since 2017. For more details see: https://stohr.sites.stanford.edu/

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Chapter 1

Introduction and Overview

Mehr Licht! (More light!) Last words of Johann Wolfgang von Goethe (1749–1832)

1.1 About the Present Book We begin with a brief outline of the motivations and objectives of the book.

1.1.1 Motivations The primary background and impetus for the present book is the development of increasingly bright sources of x-rays over the last fifty years and their use to obtain novel information about the electronic, spin, and atomic structure of matter. This development has paralleled or was interwoven with the development of optical light sources as illustrated in Fig. 1.1. While the invention of the light bulb, the discovery of x-rays, and the invention of the laser are widely recognized as revolutions, the more recent developments of new x-ray sources based on large accelerators may arguably also be associated with revolutionary developments. All five scientific revolutions shown in Fig. 1.1 involve paradigm shifts in the utilization of “light”. Within the context of the present book, we define “light” to cover the spectral range from optical to x-ray frequencies illustrated in Fig. 1.2. Light is distinguished from longer wavelength electromagnetic (EM) radiation by its quantum detection process through the photoelectric effect. The true photon nature of what we call EM radiation, however, is not restricted to the so-defined wavelength range. It is a universal feature of EM radiation despite the power of Maxwell’s classical theory of electromagnetic waves. At longer wavelengths it just becomes increasingly difficult to detect photons since typical detectors at room tem© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_1

1

2

1 Introduction and Overview

Fig. 1.1 Revolutions in the development of sources of light with time, indicated by approximate years Fig. 1.2 Definition of “light” within this book, defined through its photoelectric detection process

Spectral range of “light” x-rays

VUV

1

10

100

10 4

1000

Photon energy (eV) 10

3

10

2

1

10

10

-1

Wavelength (nm) 10

3

10

4

10

5

10

6

12

Frequency (THz =10 Hz) 10

3

10

2

1

10 -18

Cycle time (as =10

s)

perature have a thermal background noise corresponding to about 25 meV. Today, even micowave photons may be detected by use of photon counters based on Josephson junctions [1]. While the first three developments shown in Fig. 1.1 are revolutionary because of their practical impact on our everyday life, the development of the laser and the last two events are revolutionary because of the 10–15 orders of magnitude increase in photon brightness or photon density. Such increases are not just evolutionary but, using Thomas Kuhn’s definition [2], can rightfully be referred to as revolutionary. To put this development in a quantitative perspective, we show in Fig. 1.3 the number of photons produced in a birth volume of size V = λ3 by different types of

1.1 About the Present Book

3

Photon density of light sources

Photons produced in V =

Fig. 1.3 Number of photons generated in a birth volume of order λ3 , the so-called photon degeneracy parameter, for different sources [4]

3

10

10

10

=633nm

=0.1 nm

He:Ne laser

XFELs

5

1 -5

10

sun & conv. sources

SR sources

-10

10

-15

10

X-ray tubes

sources. We shall see that this number corresponds to the so-called photon degeneracy parameter or the number of first-order coherent photons [3, 4]. It is also proportional to the peak brightness of a source which is the conventional measure of its quality [5–7]. In this book we will use the terms brightness and brilliance interchangeably. The photon energy dependence or spectral distribution of the peak brightness of conventional lasers, storage rings or “synchrotrons”, and XFELs is illustrated in Fig. 1.4 [8]. Note that today the peak brightness of conventional lasers still falls off steeply beyond about 100 eV where accelerator-based sources take over.

1.1.2 Objectives The goal of the present book is to present a modern description of the nature of x-rays and their interactions with the atomic and electronic structure of matter. Building on the pioneering quantum description of radiation by Dirac [9] and Heitler [10], and modern books on x-ray science [5, 6, 11], we here extend today’s usual “semiclassical” treatment of x-rays and their interactions, based on a mixture of wave and photon concepts. The semi-classical treatment is reviewed and directly linked to that in modern quantum optics which is based on the fundamental description of light by quantum electrodynamics (QED). The quantum optics formalism, pioneered by Glauber [12–14] in the 1960s, has been widely used in the laser community [15–19], and it becomes increasingly important when the photon degeneracy parameter, or number of photons per coherence volume, is much larger than 1 as illustrated in Fig. 1.3. This is the case for XFELs where the emitted radiation contains on average a large number of photons per coherence volume. The formulation of x-ray/matter interactions through quantum optics readily allows for the simultaneous presence of many photons that may cooperate in a non-linear fashion, leading to fascinating new phenomena.

10 35

XFELs

2 2

Fig. 1.4 Peak brilliance or brightness versus photon energy of conventional lasers and higher harmonic generation sources (lasers, black envelope), synchrotron sources (blue envelope), and x-ray free electron lasers (red envelope). Figure adapted from [8]

1 Introduction and Overview

Peak brightness (photons/ s / mrad / mm / 0.1% BW )

4

10

30

10

25

Lasers Synchrotron sources 10

20

1

10

3 4 102 10 10 105 Photon energy (eV)

6

10

Our book complements and goes beyond other modern monographs on x-ray science that utilize a “semi-classical” description, such as D. Attwood’s and A. Sakdinawat’s Soft X-Rays and Extreme Ultraviolet Radiation [5] and J. Als-Nielsen’s and D. McMorrow’s Elements of Modern X-Ray Physics [6]. Mind you, the semiclassical description has served us well over the last 100 years and because of its deep roots will continue to guide our thinking in the interpretation of experiments, especially those performed with synchrotron radiation. But it is time to go beyond it. XFEL experiments utilize ultrashort and intense pulses that give rise to many -photon or non-linear effects. Many XFEL experiments today utilize femtosecond optical pulses as time references and to trigger the dynamics of electronic, spin, and structural phenomena. While x-ray science in the twentieth century has largely explored time-averaged phenomena, the new century will see increasing emphasis on dynamical phenomena which underlie the function of systems. X-ray scientists have only begun to appreciate the concepts and language of quantum optics and to apply them for the interpretation of XFEL experiments [4].1 The present book is envisioned to help bridge the existing gap in language and concepts used in the optical and x-ray laser communities which are increasing mingling.

1 It is interesting that part of the optical laser community continues to use the classical field-based formalism, reflected by the terms “strong-field” phenomena and “higher order susceptibilities”. The persistent use of this formalism may be traced to its long history and the fact that early laser experiments were also interpreted in the field-based language of nuclear magnetic resonance pioneered by Bloch, Purcell, and Rabi.

1.1 About the Present Book

5

The semi-classical description of light explains diffraction through the HuygensFresnel principle of wave interference but avoids the specification of how the diffracted intensity is detected. The crucial detection step was addressed in Dirac’s famous and bold statement that photons only interfere with themselves and not with each other [9]. Hence the conventional diffraction pattern consists of a binary assembly of 0 and 1 photon counts, corresponding to the destructive and constructive interference of a single photon wavefunction with itself. The various formulations of quantum mechanics [20] support the puzzling wave-particle duality. It is supplemented by Bohr’s complementarity principle introduced in 1935 [21] which states that in a given experiment one may only observe either the wave or photon aspect of light but not both. The acceptance of the wave-particle duality has allowed us to this day to use either the photon or wave concept to describe x-ray experiments. We typically use the wave concept to formulate interference and diffraction and the quantum concept of photons to describe atomic transitions and their spectroscopic observations. A key part of the present book is to provide a deeper understanding of when the wave and photon descriptions are equivalent and when the equivalence breaks down, i.e. when the semi-classical framework needs to be abandoned. This is elucidated here through the formulation of diffraction in a photon picture, facilitated by use of the concepts of modern quantum optics [14–16]. The validity of the semiclassical theory is based on the linearity of conventional quantum theory, which is only of first order within the more complete framework of QED [22]. Within QED, today accepted as the fundamental theory of light and matter that is valid to infinite order, light consists of elementary particles—photons. This is clearly stated by Feynman in his reader-friendly 1985 book “QED: The Strange Theory of Light and Matter” [23]. In the first order, these photons are independent and do not interact (Dirac’s statement), while with increasing order correlations between photons become allowed. The incompleteness of the wave or first order QED formulation is most beautifully demonstrated when the quantum formulation of diffraction is extended to the second order. This is a key part of our book and in the process we establish an important and remarkable new paradigm that within QED, diffraction images are direct signatures of different quantum states of light. The formulation of diffraction in a photon picture was famously introduced by Feynman in his Lectures on Physics [24], where he remarked that photon interference “has in it the heart of quantum mechanics. In reality it contains the only mystery.” The tricky aspect of the photon picture was similarly expressed in Einstein’s 1951 statement, “All the fifty years of conscious brooding have brought me no closer to the answer to the question, ‘What are light quanta?’ Nowadays every fool thinks he knows the answer, but he is deluding himself.” Feynman unified the photon with the wave description by assigning probability amplitudes to photons and electrons which closely resemble wave fields. The probability amplitude formulation becomes cumbersome with increasing order in QED [4], and for this reason we shall here utilize the more elegant albeit more abstract formulation of light and its interactions based on photon creation and annihilation operators. This formulation pioneered by Dirac [9] in the first order

6

1 Introduction and Overview

was extended to higher order in QED by Glauber [14]. We will show that the semiclassical description of light is equivalent to a first order approximation within QED, but that it breaks down in second order QED. Clearly, choices had to be made in covering different x-ray phenomena and techniques. The present book emphasizes the processes associated with “x-rays in/xrays out” in the most important forms of x-ray absorption, emission, scattering, and diffraction. We also describe the evolution of light-matter interactions with increasing energy from infrared to hard x-ray photons which naturally directs the attention to x-ray absorption edges or resonances which break up the otherwise smooth photon energy response. Due to their large cross section, they are also best suited to investigate the appearance of non-linear effects that arise in XFEL studies. Resonance phenomena form the basis of modern x-ray absorption techniques such as NEXAFS2 [29] or XANES3 and the important resonant elastic (REXS) and inelastic (RIXS) x-ray scattering techniques. Resonances also play an important role today in protein crystallography which is increasingly dominated by the use of Multiwavelength Anomalous Diffraction or MAD, where the photon energy is tuned around heavy atom absorption resonances [31, 32]. They furthermore exhibit strong polarization dependent effects which allow the separation of charge and spin phenomena in spectroscopy, microscopy, and diffractive imaging [11, 29, 33, 34]. We do not discuss extended x-ray absorption fine structure (EXAFS), which is reviewed in [25–27], details of conventional non-resonant x-ray crystallography which are well covered in the literature [6, 35, 36], and photoemission spectroscopy whose different aspects are treated in other books [37–39].

1.2 The Nature of Light: From Light Rays to QED 1.2.1 Early Concepts of the Nature of Light Over the centuries, light has been the primary tool for seeing and understanding the world and universe around us. The early belief was that light was created by a deity, like the Egyptian Sun God Ra or the biblical God who said “Let there be light”. Early theories of light were put forward by the Greek philosophers, and the concepts of light and vision remained closely linked until about 1600 when Johannes Kepler initiated a more physics-based approach to light itself [40]. While the concept of light as the opposite of darkness is fundamentally simple, the understanding of the true nature of light constitutes one of the most debated problems in the history of science. To quote Roy Glauber, who received the 2005 Nobel Prize in Physics for the development of modern quantum optics, “Few problems of physics 2

The name NEXAFS for “near edge x-ray absorption fine structure” was created to distinguish it from EXAFS [25–27] and SEXAFS [28]. 3 The name XANES for “x-ray absorption near edge structure” was coined by A. Bianconi in 1980 [30].

1.2 The Nature of Light: From Light Rays to QED

7

Fig. 1.5 Light rays falling into Grand Central Station in New York City

have received more attention in the past than those posed by the dual wave-particle properties of light” [14]. The concept of “light rays” was discussed already around 300 BC by Euclid [40]. This simple concept, based on light propagation on straight trajectories, is nicely illustrated by the photo in Fig. 1.5, showing beams of light falling through windows into Grand Central Station in New York City. This ray picture was prominently supported by Isaac Newton (1643–1727), who in 1704 put forward the view that light is corpuscular in the book Opticks: A Treatise of the Reflections, Refractions, Inflections and Colours of Light. Newton’s influence in the scientific community largely allowed the corpuscular light theory to prevail during the eighteenth century, despite challenges by Francesco Grimaldi (1618–63), Robert Hooke (1635–1703), and especially Christiaan Huygens (1629–1695) whose theory of light was published in 1690 in his book Traité de la lumiére (Treatise on Light). These scientists pointed out that the light path can deviate from rectilinear behavior, i.e. it can “diffract” around corners. Today we typically credit Huygens with introducing spherical wavefronts and Thomas Young (1773–1829) [41] and Augustin J. Fresnel (1788–1827) [42] for their use to explain diffraction around 1800.4 Fresnel also suggested the transverse nature of light and established the concepts of linear, circular, and elliptical polarization [45] which today are used to explain the magneto-optical effect discovered by Faraday in 1846 [46] and Pasteur’s discovery of chirality in 1850 [47]. 4

It was known earlier that the resolution of microscopes suffered from “diffraction effects” [43]. Credit for the formulation of the diffraction limit is typically given to Ernst Abbe (1840–1905) who in 1873 stated the minimum resolvable distance to be dmin = λ/(2 sin α), where α is the aperture’s half angle of the microscope’s objective [44].

8

1 Introduction and Overview

The central part of the wave description of light that became consolidated in the first quarter of the nineteenth century is today expressed by the magical HuygensFresnel principle, which states that every point in a light path may be considered a source of a spherical wave.5 Based on this principle, today’s diffraction theory was developed by Gustav R. Kirchhoff (1824–1887), John William Strutt (Lord Rayleigh) (1842–1919), and Arnold Sommerfeld (1868–1951) [48]. The notion of light waves was famously supported by James Clerk Maxwell’s (1831–1879) unified description of light and electromagnetic (EM) waves proposed in 1861 [49, 50]. The validity of this concept on the macroscopic length scale was proven by experiments of Heinrich Hertz around 1888 [51]. The nineteeth century concluded with two seminal developments. Around 1879, Thomas Edison (1847–1931) invented a reliable long-lasting electric light bulb.6 Today it is hard to imagine the modern industrialized world without artificial light sources, and for this reason we have listed Edison’s invention as the first modern revolution in “light” sources in Fig. 1.1. The second revolution shown Fig. 1.1 was the discovery of x-rays in 1895 by Wilhelm C. Röntgen (1845–1923) [52]. Their use has allowed us to see the invisible, from medical imaging to the atomic structure of matter. Because x-ray science constitutes the central topic of this book, we shall discuss its detailed development separately, after finishing our historical journey into the nature of light. By 1900, all observations about the behavior of light, like the phenomena of attenuation (absorption), refraction, and diffraction, could be explained by the notion that light is an electromagnetic wave.

1.2.2 The Quantum Era and Wave-Particle Duality What appeared to be a complete description of light was challenged through a paper, published on December 14, 1900, by Max Planck (1858–1947), which accounted for the spectrum of light emitted by thermal sources [53, 54]. Planck’s empirical formula described the so-called blackbody spectrum, i.e. the frequency dependent intensity distribution of light emitted by a hot gas in a closed (“black”) box through a small opening. Planck’s derivation which included a mysterious constant h = 6.55 × 10−27 [erg × sec], he termed “Naturconstante”, ushered in the quantum age. The granular nature of light emerged more directly in 1905 through Einstein’s explanation of the photoelectric effect [55], observed earlier in 1887 by Heinrich Hertz (1857–1894) [56]. Einstein introduced the idea that light is granular and that 5 It corresponds to the mathematical expansion of a plane wave as a linear combination of spherical waves. 6 Earlier efforts, including Alessandro Volta’s demonstration of a glowing wire in 1800, resulted in incandescent bulbs with extremely short lives, high production expense, and the need for high electric currents made them commercially unattractive. Edison filed for an electric lamp patent (U.S. 223,898) on November 4, 1879 which was granted on January 27, 1880. It was based on “a carbon filament or strip, coiled and connected to platina contact wires”.

1.2 The Nature of Light: From Light Rays to QED

9

such light “quanta” can transfer their energy to electrons and emit them from solids. In 1916, Millikan [57] experimentally verified Einstein’s all-important photoelectric equation through a detailed study of the photoelectric effect, measuring Planck’s constant to within 0.5%. In 1909 Einstein published a less known paper [58], where he derived a fluctuation formula for Planck’s blackbody radiation which contained a wave and a particle term. He suggested an inherent wave-particle duality in the nature of light that combines two seemingly different concepts. He concluded [59], “It is, therefore, my opinion that the next stage of the development of theoretical physics will bring us a theory of light which can be regarded as a kind of fusion of the wave theory and the emission theory.” The final breakthrough on the origin of Planck’s law came in 1916/17 through Einstein’s famous rate equation model that is based on how an atomic gas can achieve equilibrium at a given temperature through photon absorption and emission [60, 61]. In the work he introduced two types of light emission processes: spontaneous and stimulated emission.7 Historically, stimulated emission of visible light was long considered less important and it was more readily understandable since stimulated absorption and emission are completely symmetric processes for a strong classical field. It became of central importance in the development of nuclear magnetic resonance (NMR) in the 1940s and of the maser (microwave amplification by stimulated emission of radiation) and laser (light amplification by stimulated emission of radiation) in the period 1954– 1960. In practice, the emission of conventional optical sources such as light bulbs and x-ray tubes is dominated by spontaneous emission which accounts for their chaotic properties. Its introduction was the truly revolutionary part of Einstein’s paper since it could not be readily understood classically. Einstein was led to its introduction through the existence of spontaneous decays in radioactive materials. Its fundamental origin was explained by Dirac in 1927 [62, 63] as a consequence of the quantum mechanical zero-point field whose fluctuations trigger spontaneous decays.8 The complete formulation of a quantum theory of light and matter remained an elusive goal until the mid-1920s. The term “quantum mechanics” was introduced by Max Born (1882–1970) in 1924 [67] and the term for a quantum of light, the “photon”, was coined in 1926 by Gilbert N. Lewis (1875–1946) [68]. The quantum revolution occurred over the three year period 1925–1928. Although one may today distinguish nine different formulations of quantum mechanics as reviewed by Styer et 7 Einstein was quite excited about his work as evidenced by a letter to his friend Michele Besso on August 11, 1916: “Es ist mir ein prächtiges Licht über die Absorption und Emission der Strahlung aufgegangen - es wird Dich interessieren. Eine verblüffend einfache Ableitung der Planck’ schen Formel, ich möchte sagen die Ableitung. Alles ganz quantisch.” In translation “A splendid light has dawned on me about the absorption and emission of radiation—it will be of interest to you. A stunningly simple derivation of Planck’s formula, I might say the derivation. Everything completely quantized.” 8 It appears [64] that Planck first conjectured the concept of a residual energy of the radiation field [65], but it was a fuzzy concept at the time. The term “Nullpunktsenergie” or zero-point energy was phrased by A. Einstein and O. Stern in 1912 [66].

10

1 Introduction and Overview

al. [20], this period produced the best known three versions of non-relativisitic QM: Matrix mechanics by Werner Heisenberg (1901–1976), Born and Pascual Jordan (1902–1980) [69–71],9 wave mechanics by Louis de Broglie (1892–1987) [73]10 and Erwin Schrödinger (1887–1961) [76–80], and so-called transformation theory by Paul Dirac (1902–1984) [62, 63, 81, 82]. In particular, Dirac formally quantized the EM field and showed that one must treat its ground state as a dynamical system which contains the previously conjectured zero-point energy. Its associated fluctuations provided the mysterious driving force for Einstein’s spontaneous emission. In a sense, photon emission could now be perceived as amplification of the zero-point energy.

1.2.3 New Insight in the Period 1930–1950 The period from the 1930s to the 1950s saw several important developments in the description of light. In 1935 A. Einstein, B. Podolsky, and N. Rosen [83], published a famous paper, often referred to as the “EPR” paper, that outlined the “paradoxical” properties of certain two-particle states. In the same year, Schrödinger [84] coined the term “entangled states” to emphasize the fact that the properties of two quantum particles may be inextricably coupled. At the heart of the EPR paper was the authors’ conviction that it was unacceptable that for entangled states, quantum mechanics allowed the existence of a correlation between measurements made between spatially separated particles, referred to by Einstein as “spooky action at a distance”. At the time, the success of quantum mechanics, however, led most physicists to pay little attention to the EPR paper. But that changed with the development of quantum optics in the 1960s as will be discussed in this book. Another important development in the mid-1930s was the work by Pieter H. van Cittert (1889–1959) and Frits Zernike (1888–1966) resulting in the van Cittert– Zernike (VCZ) theorem [85], which allows the description of spatially incoherent light

9 Born and Jordan did not share Heisenberg’s 1932 Nobel Prize which he actually received in 1933. In 1933 Heisenberg wrote to Born, “The fact that I am to receive the Nobel Prize alone, for work done in Göttingen in collaboration—you, Jordan, and I—depresses me and I hardly know what to write to you. I am, of course, glad that our common efforts are now appreciated, and I enjoy the recollection of the beautiful time of collaboration. I also believe that all good physicists know how great was your and Jordan’s contribution to the structure of quantum mechanics—and this remains unchanged by a wrong decision from outside. Yet I myself can do nothing but thank you again for all the fine collaboration and feel a little ashamed.” [72]. 10 De Broglie first suggested the existence of a wave-particle duality not only for light but particles. He conjectured that particles are accompanied by “phase waves” [74]. The work published in more detail in 1925 [73] earned him the 1929 Nobel Prize in Physics “for his discovery of the wave nature of electrons”. De Broglie’s hypothesis was proven in 1927 by the Davisson and Germer experiment of electron diffraction by crystals [75].

1.2 The Nature of Light: From Light Rays to QED

11

emitted by thermal sources and stars.11 It laid the foundation for optical coherence theory and statistical optics, developed in more detail in the 1960s by Emil Wolf and Leonard Mandel [86, 87]. The VCZ theorem is arguably one of the most important theorems in modern optics since it characterizes the spatial coherence properties of a source. Its analogue, describing the temporal coherence, is the Wiener–Khintchine theorem [88], which was also formulated in the 1930s. In 1947 Dennis Gabor conceived the idea of holography and, by employing conventional filtered-light sources, developed the basic technique. Holography did not become commercially feasible until the advent of the laser. Another advance occurred in the 1950s through Alfred Kastler’s (1902–1984) development of optical pumping that earned him the 1966 Nobel Prize in Physics [89]. Following earlier work by A. Rubinowicz in 1918 [90] on the polarization dependence of electromagnetic transitions, Kastler’s work led to the assignment of a photon angular momentum or spin of + or − for right and left circularly polarized light, which today is widely utilized for the study of magnetic materials [34].

1.2.4 Beyond Quantum Mechanics: Quantum Electrodynamics In 1928 Dirac discovered an equation describing the motion of electrons in an external EM field [81, 82] which incorporated both the requirements of quantum theory and the theory of special relativity. It naturally led to the emergence of the electronic spin, whose interesting history has been discussed by Stöhr and Siegmann [34]. Dirac’s theory provided the foundation of quantum electrodynamics (QED), although it became soon apparent that it predicted an infinite mass of the electron [91]. Also, since each of the infinite number of oscillators in the quantum mechanical vacuum state possessed a zero-point energy, it implied that the EM vacuum had an infinite energy density and that the electron sea had an infinite negative charge density. These infinities arose from evaluating non-convergent integrals whenever corrections were attempted to Dirac’s lowest order quantum formulations. The incompleteness of Dirac’s theory became apparent by experiments in 1947/48 through the measurements of the Lamb shift of the energy levels of hydrogen by W. E. Lamb and R. C. Retherford [92] and the anomalous magnetic moment of the electron by H. M. Foley and P. Kusch [93]. The difficulties were resolved between 1948 and 50 through the diagrammatic formulations of QED by Richard P. Feynman (1918–1988), expressed by the famous Feynman diagrams, and the independently developed operator methods by Julian S. Schwinger (1918–1994) in the USA and Sin-Itiro Tomonaga (1906–1979) in Japan. All three shared the 1965 Nobel Prize in Physics since their theories were shown to be equivalent by Freeman Dyson (1923–2020) in 1949 [94]. While a complete theory of 11

Zernike received the Nobel Prize in Physics in 1953 for the development of the phase contrast microscope.

12

1 Introduction and Overview

light in the form of quantum electrodynamics (QED) now existed, its application to practical problems in optics remained a challenge. The devil remained in the details! Feynman also elucidated the equivalence of the disparate wave and particle pictures in first order QED. Using his own space-time probability amplitude formulation of QM [95], he explained Young’s double-slit experiment in his Lectures on Physics [24] by use of the particle concept for photons or electrons which, in first order QED, have the same probability amplitudes [23]. This leads to the simple picture where one may approximate a wave by summing over all probabilistic paths or probability amplitudes of a large number of independent photons. Absorption can then be viewed either as the attenuation of the incident wave or the loss of some of the photons. Diffraction can be pictured either as the scattering and interference of the incident wave or fractions of the incident photons being scattered into directions out of the beam. From a quantum perspective, Young’s double-slit experiment proceeds in single-photon processes whose probabilistic repetition leads to the buildup of the diffraction pattern [4].

1.2.5 Development of the Laser and Quantum Optics Around 1960 The ten year period around 1960 was of seminal importance for our understanding of light and consisted of several scientific breakthroughs. The best known ones are the development of the maser and laser in the late 1950s through work by Nikolay Basov, Gordon Gould, Aleksandr Prokhorov, Arthur Schawlow, and Charles Townes. While the credit for the “invention of the laser” remains controversial, the first laser was operated by Theodore Maiman on May 16, 1960. It constitutes the third revolution in Fig. 1.1. The invention of the laser, in particular, overcame the difficulty to produce strong coherent radiation in the optical wavelength region. It followed in the wake of the development of RADAR sources, short for RAdio Detection And Ranging,12 during the Second World War. The high intensity coherent microwave radiation produced by such sources led to the development of NMR in the 1940s. The late 1950s resulted in another seminal experimental and theoretical discovery of the properties of light, the Hanbury Brown–Twiss (HBT) effect [96–98]. It triggered the development of statistical optics13 in the 1960s by Emil Wolf and Leonard Mandel [86, 101] and quantum optics by Roy Glauber [12–14]. The ensuing dispute between the two camps has been chronicled by Bromberg [102]. The experiments conducted by Robert Hanbury Brown and Richard Twiss utilized thermal light either produced by a laboratory source or the star Sirius and observed 12

The term RADAR was coined in 1939 by the United States Signal Corps. Statistical optics is based on the statistical description of light, which extended the original work of Max Born [99], who together with Emil Wolf wrote their famous textbook, first published in 1959 [100].

13

1.2 The Nature of Light: From Light Rays to QED

13

a distance- or time-dependent correlation between the arrival of two photons. It was found to peak when the two photons of wavelength around 500 nm arrived at the same time and point and decreased when either their temporal separation was increased at the same point or their spatial separation was increased at the same time [4]. Remarkably, the interpretation of their results was possible within both statistical optics, where it was attributed to statistical intensity fluctuations [103] and through quantum optics, where it was interpreted as the interference of the probability amplitudes of the two photons [14, 104]. As discussed in this book, the HBT effect is a second order correlation effect in QED which for chaotic light happens to be also describable through statistical optics. We will, however, show that the quantum optics approach is more fundamental since it can account for effects, such as diffraction of entangled photons, which are outside the purview of statistical optics. In 1964 John Bell published a seminal paper [105], establishing Bell’s inequalities that allowed addressing the incompleteness of quantum mechanics suggested by the Einstein-Podolsky-Rosen paradox. Extensive laser experiments starting in the 1970s confirmed the violation of Bell’s inequalities in support of quantum mechanics. In the process, the extraordinary properties of entangled photons became the basis for the entire field of quantum information science [106]. This development has been honored by the 2022 Nobel Prize in Physics awarded to Alain Aspect, John F. Clauser, and Anton Zeilinger “for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science”. More generally, optical laser experiments have revealed the extraordinary power of the use of various aspects of light to explore the quantum world. The development is continuing to this day, owing to the enormous number of possible (quantum) states of light. There is still plenty of room for the discovery of new weird and wonderful quantum properties through optical and x-ray experiments. While the advent of conventional lasers has greatly advanced our understanding of the photon nature of light [4], the question arises whether XFELs can unveil additional aspects of “light” itself? As evidenced by the increase of Compton scattering with photon energy, the particle aspects become more pronounced for x-rays, and the intensity of XFEL beams is approaching levels where the small probability of photon-photon scattering through higher order coupling in QED can be studied [107]. At the end of the historical journey through the understanding of light, we summarize in Fig. 1.6 key milestones associated with our discussion above. These milestones in the understanding of optical light are discussed in more detail in [40, 108, 109]. We shall extend this picture to x-rays in the following sections.

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1 Introduction and Overview

Modern milestones in production/nature of visible light Zernike Gabor 1934-1938 1947

Huygens 1690

Young 1801

Newton 1704

Fresnel 1818

1700

Maxwell 1861 Kirchhoff 1883 Hertz 1888 Sommerfeld 1896 Rayleigh 1897 Edison 1879

1800

Light bulb

Basov Gould de Broglie Maiman Born Prokhorov Dirac Feynman Schawlow Einstein Heisenberg Schwinger Townes Tomonaga 1954-1960 1905, Jordan Hanbury Brown Planck 1909, Schrödinger Dyson 1925-1928 1948-1950 Twiss 1956 1900 1916

1900

1925

1950

Black-body rad. QM QED PE effect “photon” Coherence theory duality Holography

Glauber 1963 Mandel, Wolf 1965

1960

Maser Laser HBT effect

Fig. 1.6 Important milestones from around 1700 to about 1965 associated with the production of visible light (blue) and theoretical and experimental advances in understating the nature of light (black)

1.3 The X-Ray Revolution 1.3.1 The Discovery and Early Utilization of X-Rays The second modern revolution in the production of what we have defined as “light” was the discovery of x-rays in 1895 by Wilhelm C. Röntgen (1845–1923) [52], as indicated in Fig. 1.1. Because of the central role of x-ray science in this book, we shall discuss its development now that we have completed the historical journey associated with visible light. In our discussion we can of course benefit from the previous discussion of visible light which is only distinguished from x-rays by the photon energy or wavelength as shown in Fig. 1.2. Röntgen’s work was immediately accepted as a revolutionary discovery by both scientists and society at large. Only six years later he was awarded the first Nobel Prize in Physics “in recognition of the extraordinary services he has rendered by the discovery of the remarkable rays subsequently named after him.” Röntgen referred to the radiation discovered by him as “X-Strahlen” (x-rays), with “X” reflecting their unknown character and “rays” because of the shadow images they formed of bone and other dense objects. In German and many other languages, the “X” is replaced by “Röntgen” in honor of their inventor. In further experiments, Röntgen did not succeed in observing their refraction by prisms, their focusing by lenses or their deflection by magnetic fields. He concluded that x-rays were not susceptible to conventional refraction and reflection and were distinct from Lenard’s cathode rays, which were shown to be electrons by J. J. Thomson in 1897. The early developments in x-ray science are partly chronicled by numerous Nobel Prizes, summarized in Fig. 1.7 [110]. The early Nobel Prizes were awarded in Physics

1.3 The X-Ray Revolution

15

Nobel Prizes based on the Utilization of X-Rays Physics

Chemistry

1901: WILHELM RÖNTGEN

1936: PETER DEBYE

1914: MAX VON LAUE

1962: MAX PERUTZ and SIR JOHN KENDREW

1915: SIR WILLIAM HENRY BRAGG and SIR WILLIAM LAWRENCE BRAGG

1964: DOROTHY HODGKIN 1976: WILLIAM LIPSCOMB

1917: CHARLES BARKLA

1985: HERBERT HAUPTMAN and JEROME KARLE 1924: KARL MANNE SIEGBAHN 1927: ARTHUR COMPTON

1988: JOHANN DEISENHOFER, ROBERT HUBER and HARTMUT MICHEL

1981: KAI SIEGBAHN

1997: PAUL D. BOYER and JOHN E. WALKER

Medicine

2003: PETER AGRE, and RODERICK MACKINNON 2006: ROGER KORNBERG

1946: HERMANN JOSEPH MULLER 1962: FRANCIS CRICK, JAMES WATSON and MAURICE WILKINS

2009: VENKATRAMAN RAMAKRISHNAN, THOMAS STEITZ and ADA YONATH 2012: ROBERT LEFKOWITZ and BRIAN KOBILKA

1979: ALAN M. CORMACK and SIR GODFREY N. HOUNSFIELD

Fig. 1.7 List of Nobel Prizes associated with the nature and applications of x-rays

for contributions to the understanding of x-rays and their interactions with matter. Later Nobel Prizes in Chemistry and Medicine are typically associated with pioneering structure determinations of macromolecules such as proteins that are used to imply their function. While the use of x-rays for medical imaging was demonstrated by Röntgen in the process of their discovery, their interactions with matter in the form of atoms, molecules, and solids began to be investigated by others at the dawn of the twentieth century. The work focused on three fundamental types of the x-ray responses: absorption, emission, and scattering. They are also the key processes discussed in this book. X-ray absorption and emission spectroscopy was pioneered by the English physicist Charles G. Barkla (1877–1944), who in 1905 found that x-rays are polarized perpendicular to the direction of propagation [111] and thus must be transverse waves.14 He also showed that each element contained characteristic x-ray absorption thresholds [112] and first distinguished the primary x-ray emission lines, labelling them the K and L series [113]. The work, for which he won the 1917 Nobel Prize in Physics, is summarized in his 1918 paper [114]. The characteristic change of the photon energies of the emitted K lines was studied in more detail in 1913/1914 by Henry G. J. Moseley (1887–1915) in England, leading 14

Interestingly, the Braggs were Barkla’s vociferous opponents since they originally thought that γ - and x-rays were a neutral pair of particles, an electron coupled to a positively charged particle.

16

1 Introduction and Overview

Fig. 1.8 Moseley’s photographic recording of Kα and Kβ emission lines for different 3D elements and brass (an alloy of Cu and Zn) [116]

Wavelength

to Moseley’s law [115–117], which states that the K emission energies scale with the square of the atomic numbers. It also provided a practical means of using xray spectroscopy to determine atomic composition of samples and their order in Mendeleev’s periodic table. Moseley’s observation of the dependence of the atomic emission lines for different 3D transition metals is shown in Fig. 1.8. Moseley’s untimely death in 1915 during the battle of Gallipoli at age 27 likely cost him the Nobel Prize. This is evidenced by the fact that two Nobel Prizes in Physics for x-ray spectroscopy were later given to Barkla in 1917 and to Swedish physicist Manne Siegbahn (1886–1978) in 1924. Manne Siegbahn began his studies of x-ray spectroscopy in 1914 and initially used the same type of spectrometer as Moseley. He then developed improved spectrometers to make very accurate measurements of the x-ray wavelengths emitted by different atoms [118, 119], resulting in an almost complete understanding of the electron shell [120]. Barkla’s early studies of the x-ray absorption fine structure of matter were continued in more detail in 1918 by Wilhelm Stenström and Hugo Fricke in Manne Siegbahn’s laboratory, and an early theoretical interpretation was put forward by Walter Kossel in 1920. Significant progress in its interpretation was however only made after the development of quantum theory when in 1931 Ralf de L. Kronig (1904–1995) published a seminal paper [121]. It may be considered the founding theoretical paper of modern x-ray absorption spectroscopy. The interesting history of x-ray absorption spectroscopy has been reviewed by Christian Brouder under the pseudonym R. Stumm von Bordwehr15 [122] and by Lytle [123]. 15

General Stumm von Bordwehr is a comical figure, “the man without qualities”, in a novel by Robert Musil.

1.3 The X-Ray Revolution

17

The early spectroscopic x-ray absorption and emission studies were complemented by studies of the diffractive properties of x-rays. On June 8, 1912, Paul Knipping and Walter Friedrich, who were assistants of Max von Laue (1879–1960), first reported the diffraction of x-rays by crystals, and the results were published in 1913 by W. Friedrich, P. Knipping and M. Laue [124], with Laue writing the theoretical and the two others the experimental part of the paper. Only a year later, Laue received the 1914 Nobel Prize in Physics “for his discovery of the diffraction of x-rays by crystals.” The work in Germany was immediately followed up in England by William Henry Bragg (1862–1942) and William Lawrence Bragg (1890–1971). Bragg’s law nλ = 2d sin θ was first presented on November 11, 1912, to the Cambridge Philosophical Society and published in 1913 [125]. In the following year, Paul Peter Ewald (1888–1985) devised a graphic method of solving the Bragg equation for crystals, now known as Ewald’s sphere, where the scattering vector, corresponding to the momentum transfer, is equal to a reciprocal lattice vector [126]. The Braggs were awarded the 1915 Nobel Prize in Physics. The work of von Laue, the Braggs, and Ewald gave birth to the field of x-ray crystallography that has revolutionized our understanding of the atomic structure of all forms of matter. The particle-like nature of light suggested by Einstein in 1905 was supported in 1922 by an experiment carried out by Arthur H. Compton (1892–1962). He studied the scattering of x-rays from electrons in a carbon target and found that the scattered x-rays had a longer wavelength than the incident ones, and that the shift in wavelength increased with scattering angle [127]. Compton explained and modeled the data by assuming a particle (photon) nature for light and applying conservation of energy and momentum to the collision between the photon and the electron. Historically, Einstein’s photoemission and Compton’s scattering effects were widely accepted as proof that light cannot be explained purely as a wave phenomenon.16 Compton’s measurement and explanation earned him a share of the Nobel Prize in physics in 1927.

1.3.2 Development of Synchrotron Radiation Sources The first x-ray tubes emitted relatively weak intensities consisting of atomic emission lines and a broad bremsstrahlung background. After J. J. Thomson’s (1856–1940) discovery of the fact that “cathode rays” were negatively charged “electrons” in 1897, Emil Wiechert (1861–1928) [130] and Alfred-Marie Liénard (1869–1958) independently worked out what is now called the Liénard-Wiechert potentials of a moving charge [131, 132]. In particular, Liénard’s paper of 1898 already contained a 16

It was later realized that both effects can be explained by assuming classical electromagnetic waves with quantum properties associated only with the atomic and electronic building blocks of matter [128]. Today, the experiment performed in 1986 by Grangier, Roger, and Aspect [129] is often considered as additional proof for the existence of photons.

18

1 Introduction and Overview

Fig. 1.9 First synchrotron light emitted by the 70 MeV synchrotron at general electric. The light is visible as a bright spot below the center of the picture on the left [134]

basic theory of synchrotron radiation. It did not require use of the theory of relativity because Maxwell’s equations are invariant under transformations between moving systems. In 1912 the theory of synchrotron radiation was worked out in more detail by the mathematician G. A. Schott [133]. The radiation could not be observed, however, until circular electron accelerators became available. Synchrotron radiation was first observed in 1947 at General Electric Research Laboratory on a 70 MeV synchrotron [134], as shown in Fig. 1.9. In the 1960s the use of synchrotron radiation for the study of matter began at synchrotrons in Washington, D.C (USA), Tokyo (Japan), Hamburg (Germany), and Frascati (Italy). In accelerator-based sources the kinetic energy of electrons is increased from tens of keV in conventional x-ray tubes to energies of the order of a GeV, surpassing the relativistic limit set by the rest mass of the electron, 511 keV. The relativistic reduction of the Coulomb repulsion between the electrons then gives rise to the emission of radiation into a narrow forward cone when the electrons are bent by a magnetic field. In 1968, first experiments done on an electron storage ring located at the University of Wisconsin in Madison, named Tantalus I, revealed the superiority of storage rings over synchrotrons. In a storage ring, the electron bunches circulate at a fixed energy at a high repetition rate of about 500 MHz for periods up to many hours, providing a stable broadband radiation spectrum of high intensity and stability. Despite the nearly exclusive use of storage rings for the production of x-rays today, the historical name “synchrotron radiation” (SR) has been maintained. Today, one distinguishes three stages of further development after the start of the “synchrotron revolution” around 1970 [5–7]. The first generation of storage

1.3 The X-Ray Revolution

19

ring sources was primarily operated for high energy physics and the by-product of synchrotron radiation was used parasitically. The second generation utilized storage rings that were operated specifically for the production of x-rays. The third generation of sources was specifically designed for optimizing the radiation quality or spectral brightness of the source and included the use of undulators and wigglers.17 One may roughly associate the first generation with the years < 1980 with third generation sources taking over after about 1990. The first and second generation of synchrotron radiation (SR) experiments mainly benefitted from the large number of electrons Ne ∼ 1010 that could be concentrated in individual bunches due to relativistic effects and the high pulse repetition rate R ∼ 500 MHz that could be achieved by the use of storage rings. The average flux, , or number of photons per unit time per unit area, emitted by such sources, simply scales as  ∝ R Ne . The reason for this scaling is that the fields of the individual electrons in the centimeter sized bunches do not act as a single macro-charge but rather add incoherently. In third generation synchrotron sources the electron bunch size was optimized for x-ray emission by arrangement of electron optical elements around the ring, and undulators were used to enhance the brightness by periodically wiggling the electron bunch. This increased the average brightness relative to a single arc bending magnet source by a factor n 2u ∼ 104 , where n u ∼ 100 is the number of undulator periods. In the process, the broad bandwidth bending magnet spectrum becomes concentrated into a narrow bandwidth around a selected photon energy. The factor of 104 enhancement in average brightness became the driving force for the construction of numerous third generation light sources around the world. It is continued today by the quest to build “ultimate” storage rings where the emission is limited only by the quantum mechanical space-momentum uncertainty relation [7, 137].

1.3.3 The Advent of X-Ray Free Electron Lasers The final revolution in x-ray science in Fig. 1.1, caused by the construction and use of XFELs, happened at the dawn of the twenty-first century. The foundation for this development was laid in the 1970s by John Madey and coworkers at Stanford University, USA [138–140]. The concept was based on creating a radiation cavity by use of mirrors that reflected the radiation back and forth across a straight section in a storage ring, similar to a conventional laser cavity. In the 1980s Madey’s concepts were extended and the mirrors which limited the spectral range were omitted. The novel concept which allowed the generation of radiation up into the hard x-ray regime was based on creating a self-ordering effect in a single pass of a small electron bunch through a long undulator, a process called self-amplified spontaneous emission (SASE), owing to the fact that it resembles the 17

The term brightness and its significance were introduced for the description of SR by K.-J. Kim in 1986 [135] following an earlier paper that focused on x-ray coherence [136].

20

1 Introduction and Overview

LCLS at SLAC/Stanford

injector

undular hall ~140 m

1km linac electron beam

undulator hall

near-hall stations

x-ray beam

far-hall stations

Fig. 1.10 Schematic of the Linac Coherent Light Source (LCLS) XFEL superimposed on an arial photograph of SLAC National Accelerator Laboratory at Stanford University, California. On the right is a picture of the long undulator hall where the x-rays are produced

amplification of spontanaeous emission (ASE) in optical lasers. The process was first discussed by the Russian scientists A. Kondratenko and E. Saldin [141, 142]. Shortly thereafter, the theory was solidified by the Italian scientists R. Bonifacio, C. Pellegrini, and L. Narducci [143], who clarified the fundamental physics and presented a simple model based on a single FEL parameter that depends on the electron beam density, its energy, the undulator period, and its magnetic field. The detailed historical development of FELs has been reviewed by Pellegrini [144], and its modern theory is given in [7, 145]. The first FEL facility built for user operation was the SASE VUV/soft x-ray facility FLASH (Free electron LASer Hamburg) at DESY in Hamburg, Germany, which began operation in 2005. It utilized a superconducting linac, covering the vacuum ultraviolet spectral range ω < 100 eV. In 1992 Claudio Pellegrini presented a concrete plan of building a SASE-based “hard x-ray” FEL (XFEL) using the existing 3 -km-long linac at SLAC/Stanford equipped with a high-quality photoinjector [146]. This later led to the Linac Coherent Light Source (LCLS) which utilized the last 1 km of an existing linac, providing an electron beam energy 4–14 GeV that covered the spectral range of about 300 eV– 10 keV with 120 pulses/s. LCLS started its user program in 2009 [147]. It is currently being upgraded (LCLS-II) by adding a superconducting accelerator structure to the first km of the linac which provides 106 pulses/s. An areal view of the large-scale facility which began operation in 2009 and a photograph of the undulator hall where the photons are created are shown in Fig. 1.10.

1.3 The X-Ray Revolution 2

10

10

ng2Ne

21

XFEL, gain length, ng ~ 100 periods

Number of photons per pulse

/ ~

~1 fs

4 c

8

Ne~10 electrons

transform limited pulse

nu2Ne

undulator, nu ~ 100 periods 1

10

/ ~ nu

5

~100 ps

Ne

bending magnet

/ ~1 1 0.1

10

Ne~10 electrons

C

1 10 Photon energy (keV)

100

Fig. 1.11 Comparison of the spectral distribution and photons per pulse for a bending magnet and undulator source in a storage ring with an ideal XFEL, emitting diffraction, and transform-limited pulses. We assumed Ne = 1010 electrons per bunch for the storage ring, corresponding to a charge of 1.6 nC, a beam energy Ee = 3 GeV, and an undulator length of n u  100 periods. For the XFEL we assumed the same beam energy, a gain length of n g  n u  100 periods, and that the Ne = 108 electrons per bunch are ordered in sheets to produce coherent pulses of about 1 fs duration

During the 1990s the XFEL concept was further developed by accelerator scientists. The revolutionary evolution of the emitted x-ray spectrum from bending magnets to undulators in storage rings to a spatially and temporally coherent XFEL is illustrated in Fig. 1.11. In the figure we have assumed the production of a single coherent x-ray pulse of about 1 fs duration, which is a typical length of a single spike in a SASE pulse [4, 148]. Linacs have the advantage over storage rings that a better spatial and temporal compression of electron bunches can be achieved. The bunches are used only once before they are discarded. For example, an electron bunch compressed to a length of = 3µm produces an x-ray pulse of the same length corresponding to τ = /c = 10 fs (1 fs=10−15 s). More importantly, when the electron bunches are sent through of a long undulator, regions of the bunches self-order in the field of the spontaneously emitted synchrotron radiation into sheets separated by the wavelength. The in-phase emission of the ordered electron sheets then results in a coherent radiation enhancement, which scales with the square of the number of electrons Ne2 in the ordered regions. For short bunch lengths of about 1 fs that may contain Ne  108 electrons, the number of photons per pulse is about 1010 as shown in Fig. 1.11.

22

1 Introduction and Overview

Nature

Technology -3

hummingbird wing motion ~ 0.1 ms

10 s

1 ms camera shutter speed ~ 130 s flash ~ 30 s

protein folding ~ 10 s

-6

10 s

molecular group motion ~ 1 ns

-9

10 s

1 s

1 ns

Magnetic recording time per bit ~ 1 ns Computing time per bit ~ 100 ps

10

-12

s

1 ps

ultrafast technology void

atoms oscillate in ~ 100 fs

atomic electron circles in ~ 1 fs

10

-15

s

1 fs

Fig. 1.12 Timescales involved in processes in nature and technology. One may obtain an approximate time duration t of a process by considering the energy E involved by use of the relation t E  , where  = 0.66 fs eV

To take advantage of the extraordinary XFEL capabilities, x-ray scientists began to develop a convincing scientific case in various scientific disciplines, required for funding XFEL construction. The early scientific case was largely based on the use of high intensity ultrashort pulses which for the first time opened the door for recording movies of atomic motions [149]. The femtosecond XFEL pulses are faster than the timescale of atomic motion which proceeds with the speed of sound of about 1 nanometer per picosecond (1 ps = 10−12 s). They may even capture charge and spin dynamics during the creation and dissociation of chemical bonds. The timescales of various motions in nature and technology are illustrated in Fig. 1.12. They follow the rule of thumb “the smaller the faster”, expected from the concept of inertia (Newton’s first law). In 2010, the FERMI FEL in Trieste, Italy, which constituted a seeded version of a single-pass high-gain FEL, became operational. Seeded operation provides improved control of the FEL spectrum, pulse duration, and synchronization with conventional laser sources for pump and probe applications. An initial seed pulse, provided by a conventional, high peak power, pulsed laser operating at 260 nm is temporally

1.3 The X-Ray Revolution Fig. 1.13 Timeline of developments of free electron lasers covering the vacuum ultra violet (VUV) to hard x-ray spectral range. Colors indicate presently covered spectral range

23 Plans for a VUV FEL Los Alamos (USA)

1988

Proposal to build an XFEL at SLAC (USA)

1992

Photon energy VUV/ Soft x-ray Hard x-ray

2005

LCLS (USA) begins user operation

2009

FERMI@Elettra (Italy) begins user operation

2011 2012

FLASH (Germany) begins user operation

SACLA (Japan) begins user operation

PAL-XFEL (South Korea) begins user operation

Completion of the DCLS VUV FEL (China)

LCLS-II (USA) expected to begin operation

2017

European XFEL (Germany) begins user operation

2019

SwissFEL (Switzerland) begins user operation

2021

SXFEL (China) first experiments

2023 2025

SHINE (China) expected to begin operation

synchronized with and superimposed on electron bunches from a linac, causing a spatial density modulation at the seed wavelength that also contains higher harmonics. One of the higher harmonics in the vacuum ultraviolet (originally up to 20 nm, later extended to about 4 nm) is then amplified by an undulator to a high peak power level. The temporal duration of the FEL radiation is approximately that of the seed laser (about 100 fs). The upconversion process used in FERMI becomes increasingly difficult with increasing photon energy, and SASE XFELs therefore typically use a variant called self-seeding. A broadband SASE photon pulse is first sent through a monochromator and is then again superimposed on the original electron bunch for amplification [150, 151]. Figure 1.13 summarizes the historical development of free electron lasers in the vacuum ultra violet (VUV) to hard x-ray regions around the world. The presently covered spectral range is indicated in color, which in some cases has been extended after the original construction.

24

1 Introduction and Overview

Fig. 1.14 Growth of protein structure determinations per year by x-ray crystallography, NMR, and electron microscopy deposited into protein data bank. Figure courtesy of Anton Barty and Henry Chapman, DESY

1.4 The Scientific Power of X-Rays 1.4.1 From Optical to X-Ray Response of Matter Today, x-rays are well known for medical imaging and for their ability to reveal the atomic structure of matter owing to their short wavelength. X-ray diffraction from the atomic building blocks of matter has been responsible for some of the great breakthroughs during the twentieth century: the understanding of the atomic structure of materials and the fundamental worker molecules of life, proteins. The remarkable increase of x-ray structure determinations of proteins and the dominant use of x-rays are illustrated in Fig. 1.14. The process responsible for diffraction is elastic scattering, corresponding to wiggling of the total atomic electron clouds followed by re-emission into random directions. It is referred to as Rayleigh scattering in the optical and Thomson scattering in the x-ray regions. X-ray diffraction in the form of Bragg diffraction results from coherent addition or constructive interference of Thomson scattering into specific directions. Today, the determination of de novo crystal structures of biological macromolecules is naturally dominated by the utilization of synchrotron and XFEL sources due to their high intensity. Importantly, however, it is the tunability of the x-ray energy to absorption edges of heavy embedded atoms, like the Se K-edge at 12.6 keV, which underlies the majority of protein structure determinations [32]. The

1.4 The Scientific Power of X-Rays 8

10

Fe 6

10 Cross Sections (barn/atom)

Fig. 1.15 Interaction cross sections of photons with Fe atoms as a function of photon energy extending from the optical to the x-ray region (1 barn = 10−10 nm2 )

25

Absorption 4

10

Elastic scattering 2

10

1

0.01 1

K M2,3

L2,3

10 100 1000 Photon Energy (eV)

10

4

so-called Single-wavelength Anomalous Diffraction (SAD) and Multi-wavelength Anomalous Diffraction (MAD) methods facilitate the solution of the “phase problem” by exploiting the “anomalous” changes in the scattering phase near the absorption edge [31, 32, 152]. The changes of the scattering and absorption cross sections from the visible to the x-ray range, and the dramatic resonance effects at absorption “edges”, are illustrated in Fig. 1.15 for Fe atoms in the elementary metal. The x-ray absorption cross section is composed of contributions from the individual atomic shells. In the optical range it is determined by transitions within the outermost valence shell, and with increasing photon energy, the additional excitation of electrons from inner shells, leading to the emission of photoelectrons, gives rise to absorption steps or “edges”. At photon energies above about 100 eV the absorption cross section decreases, accounting for deeper penetration of x-rays into materials, whereas the non-resonant Thomson scattering cross section levels off and remains approximately constant. With increasing photon energy one can liberate electrons from solids by overcoming the surface potential or workfunction, utilized in ultraviolet photoelectron spectroscopy (UPS), or at higher photon energies eject core electrons in x-ray photo electron spectroscopy (XPS) [37]. The excitation of inner shell electrons by x-rays leads to another fundamental difference between the absorption and emission of optical and x-ray photons. The created core holes may be filled by outer shell electrons, with the electronic decay resulting in the emission of photons as well as Auger electrons. The Auger decay channel actually dominates for the 1s or K-shell excitation of atoms lighter than Mn (Z = 25) and for nearly all atoms for 2 p or L-shell excitation. More details of characteristic changes of absorption from the infrared to the x-ray region are shown in Fig. 1.16 for liquid water [153–155], represented by the polarization averaged linear absorption coefficient μx . The lowest infrared to optical region

26

(nm -1 )

10

-1

H2O water

O K-edge

-3

x

10

10

-5

10

-7

10

1.6

Cross section (Mb)

Absorption coefficient

Fig. 1.16 Absorption coefficient μx of liquid water over a large photon energy range [153, 154]. The inset shows the enlarged O K-edge fine structure [155], plotted as the absorption cross section per atom σabs (1 Mb = 10−4 nm2 )

1 Introduction and Overview

-9

10

1.2 1 0.8 0.6 0.4 0.2 0 530

-11

1

0.1

O K-edge finestructure

1.4

535

540

545

102

10

Photon energy (eV)

10 3

10 4

550

Photon energy (eV)

10 5

555

560

103

10 6

104

10 7

10 8

Photon energy (cm -1)

10

2

10

3

10

4

10

5

10

6

Photon energy (THz)

10

4

10

3

10 2

10

1

0.1

Wavelength (nm)

reveals detailed structure arising from rotational, vibrational, and electronic excitations. The x-ray response above 100 eV is considerably simpler and dominated by the O K-edge absorption increase near 540 eV. Closer inspection of the K-edge reveals the fine structure shown enlarged in the inset below [155]. It arises from transitions of O 1s core electrons to unoccupied molecular states of the H2 O molecules [29] with modifications due to H-H interactions in water [155]. Another example is given in Fig. 1.17, where the absorption response of Co metal and the insulator CoO are compared. The data have been synthesized over a large energy range from various sources [154, 156–161]. In the optical region, absorption in the insulator CoO is suppressed by the existence of a band gap of  2.5 eV, while Co metal shows a rather smooth response. Above 100 eV, the x-ray response differs at the positions of the oxygen and cobalt absorption resonances, as indicated. The fine structure near the Co L3 absorption threshold, shown in the inset, reflects the difference in the local bonding configurations in the two cases [33, 34]. While the characteristic atomic absorption “edges” lead to atomic specificity, the fine structure near these “edges”, referred to as near edge x-ray absorption fine structure (NEXAFS) or x-ray absorption near edge structure (XANES), provides information on chemical bonding or coordination of the absorbing atom.

10

-1

10

-2

10

-3

10

-4

10

-5

x

Absorption coefficient

Fig. 1.17 Frequency dependence of the x-ray absorption coefficient μx for Co metal [154, 157, 158, 160] (red) and CoO [154, 156, 159, 161] (black) over a wide photon energy range. Note in particular the strong difference of absorption in the optical region due to the 2.5 eV band gap in CoO

27

(nm -1 )

1.4 The Scientific Power of X-Rays

Co M-edge

Co metal

band gap

CoO L3 resonant structure

776

1

Co L-edge

778

10

780

O K-edge

Co K-edge

782

100 1000 Photon energy (eV)

10

4

1.4.2 The Importance of X-Ray Resonances The importance of x-ray resonances, which occur at atomic absorption thresholds (“edges”), is based on the following key properties: (1) their elemental or atomic specificity, (2) their enhanced cross sections, (3) their chemical or bonding sensitivity, and (4) their polarization or orientational dependence. Owing to the short x-ray wavelength these properties may be utilized to obtain nanoscale structural resolution through spectro-microscopy. While the elemental specificity and enhanced cross sections of the resonances are apparent from Fig. 1.17, we shall briefly demonstrate the chemical and polarization effects.

1.4.2.1

Chemical or Bonding Specificity

The chemical information regarding local bonding is beautifully illustrated in Fig. 1.18 for different covalent bonds of selected C atoms with its C, N, and O neighbors [162, 163]. In the shown molecular complexes, central C atoms form bonds with different neighbor atoms through the formation of molecular orbitals. One typically distinguishes π and σ molecular orbitals which are respectively oriented perpendicular and parallel to the internuclear axis [29]. In x-ray absorption, one observes the transitions of C 1s core electrons into the empty parts of these orbitals, commonly denoted as π ∗ and σ ∗ orbitals. The different empty orbitals are then directly revealed by different resonances in the x-ray absorption spectra. The splitting and shifting of resonances arise from both chemical core level shifts as observed in x-ray photoelectron spectroscopy (XPS) as well as from the splitting of the empty valence orbitals [29]. The change in energy position and intensity

28

1 Introduction and Overview

Fig. 1.18 Near edge resonances of carbon atoms that are covalently bonded to C, N, and O atoms, revealing the sensitivity of resonance intensities and peak positions to chemical environments [162, 163]

C K-edge

C

C

Polyvinyl methylketone

C

O

N

C

O

C-H

O

Absorption intensity

C

C

Nylon-6

C

O

Polymethyl Methacrylate N

C

N

O

Polyurea O

C

N

O

Polyurethane N

O

C

O

O

O

Polycarbonate

284

286 288 290 Photon Energy (eV)

292

of different resonances may be used to study changes in molecular bonding upon chemical transformations in various forms, for example in molecule-surface reactions [29, 164] or even in transient reaction intermediates on ultrafast time-scales [165, 166].

1.4.2.2

Polarization Dependent Spectroscopy

The importance of the polarization dependence of the resonances in x-ray absorption spectra is illustrated in Fig. 1.19. One typically distinguishes the limiting cases of linear and circular polarization. In linearly polarized light, the electric field vector which defines the polarization direction oscillates in magnitude along an axis. When interacting with a sample, the polarization vector acts as a search light for empty valence orbitals oriented along the

1.4 The Scientific Power of X-Rays

Absorption Intensity

X-ray Natural Linear Dichroism

*

(a)

C6H6 /Ag(110) C K edge

E

*

285 305 Photon Energy (eV)

325

X-ray Magnetic Linear Dichroism Absorption Intensity

Fig. 1.19 a X-ray natural linear dichroism illustrated by the polarization dependence of the C K-edge absorption spectrum of benzene (C6 H6 ) bonded to a Ag(110) surface, revealing that the molecule lies down on the surface [167]. b X-ray magnetic linear dichroism of the Fe L2 resonances for a LaFeO3 film due to the preferential orientation of the antiferromagnetic axis [168]. c X-ray magnetic circular dichroism for a ferromagnetic Fe film, revealing the direction and size of the magnetic moments [160]

29

LaFeO3 Fe L2 edge

(b)

720

E

724 722 Photon Energy (eV)

Absorption Intensity

X-ray Magnetic Circular Dichroism

(c)

L3

Fe metal Fe L edges

L

L2

700

720 Photon Energy (eV)

740

axis. Parallel alignment maximizes the transition intensity, so that the polarization dependence directly determines the valence orbital orientation in the sample [29, 164]. The so-called x-ray natural linear dichroism (XNLD) effect is illustrated in Fig. 1.19a for benzene molecules bonded to a Ag(110) surface [167]. Polarized xray absorption spectroscopy may even detect the existence of a minute orientational order in structurally highly disordered systems [169–172]. The related effect of x-ray magnetic linear dichroism (XMLD) exists only in the presence of a magnetic alignment axis in a sample [34, 173–175]. It arises from the spin-orbit coupling which leads to a distortion of the charge density relative to the preferred spin orientation axis. The charge asymmetry then causes a difference in the x-ray absorption intensity when the polarization vector is parallel and perpendicular to the magnetic axis. XMLD is often used today for the determination of the orientation of the antiferromagnetic axis in thin films as illustrated in Fig. 1.19b [168].

30

1 Introduction and Overview

Circular polarization corresponds to the temporal clockwise (right) or counterclockwise (left) rotation motion of the electric field at a fixed point along the propagation direction. This rotation in time leads to a lack of time reversal symmetry and the existence of a photon angular momentum. The orientation of this photon angular momentum relative to an angular momentum or magnetic moment in a sample then leads to a prominent x-ray magnetic circular dichroism (XMCD) effect [33, 34, 176–179], as shown in Fig. 1.19c [160]. If one takes a snapshot in time, the polarization vector of the light wave shows a helical motion in space along the propagation direction creating a lack of inversion symmetry. When the bonding around the absorbing atom also lacks a center of inversion or exhibits a handedness in space, the relative alignments of the two helicities give rise to the x-ray natural circular dichroism (XNCD) effect [180–187]. In a molecular orbital picture a handedness of the charge density arises from the mixing of p and d orbitals by the ligand field and this “twist” or handedness in the valence orbitals is sensed by the handed electric field. This effect is typically small and does not exist in the dipole approximation. Today, soft x-ray absorption studies may also be performed with table-top higher harmonic laser sources [188, 189]. In particular, polarization and time-dependent studies using the M-edges of the 3d transition metals Fe, Co, and Ni, which occur below 100 eV as shown for Co in Fig. 1.17, have emerged as a convenient way to study element-specific femtosecond magnetization dynamics [190, 191]. Polarization dependent studies are also of importance in x-ray emission spectroscopy (XES). Today, x-ray emission studies are typically carried out after resonant excitation which enhances the signal. It is then referred to as resonant inelastic x-ray scattering or RIXS. While RIXS is in general a complicated process [192– 197], the process is quite simple for atoms and molecules bonded to surfaces [164, 198, 199]. As reviewed by Nilsson and Pettersson [199], RIXS has provided detailed information on the nature of molecular bonding to surfaces which plays a key role in heterogeneous catalysis [199–201]. As an example we show in Fig. 1.20 C K-shell absorption and emission spectra of oriented benzene C6 H6 molecules on Cu(110) and Ni(100) surfaces [202, 203]. For benzene, the π (perpendicular to molecular plane) and σ (in plane) orbitals can be separately observed through the polarization dependence of the transitions in absorption and emission. Resonances in the x-ray absorption spectra shown on the right of Fig. 1.20a correspond to transitions of a 1s core electron into the lowest unoccupied molecular orbitals of benzene which are anti bonding in nature. The polarization dependence is due to the alignment of the incident linear polarization (electric field) vector relative to the spatial direction of molecular orbitals [29], as previously shown in Fig. 1.19a. In contrast, XES or RIXS spectra reflect transitions from filled valence states, typically responsible for bonding, into the core hole created in the x-ray absorption step. In RIXS, illustrated on the left of Fig. 1.20, a polarization dependence arises from the transverse nature of the emitted light with the polarization vector lying in a plane perpendicular to the photon emission direction [198]. For emission normal to the surface, shown in blue, only σ orbitals are observed. At glancing emission from

1.4 The Scientific Power of X-Rays -20

31 -10

(a)

0

0

10

C6H 6 /Cu(110)

X-ray absorption

Intensity

X-ray emission

*

260 280 270 Emission energy (eV) -20

(b)

20

-10

0

*

290 300 Incident energy (eV) 0

10

20

C6H 6 /Ni(100)

260 270 280 Emission energy (eV)

290 300 Incident energy (eV)

Fig. 1.20 a Illustration of the complementarity of polarization dependent C 1s ↔ 2 p x-ray absorption and emission spectra for benzene (C6 H6 ) molecules bonded in a lying down configuration to a Cu(110) surface [202, 203]. In x-ray absorption (right) the empty valence orbitals aligned parallel to the electric field polarization vector are revealed by the spectra, as shown on the right. In x-ray emission (left), the probed orbitals are determined by the emission direction. In the shown red spectrum the parallel component has been eliminated. b Complementary spectra for C6 H6 bonded to a Ni(100) surface [202], revealing increased surface chemical bonding through changes of resonances in position and intensity

the surface, both perpendicular (π) and parallel (σ ) bonds to the surface are observed. In the shown red emission spectrum on the left of Fig. 1.20, the contributions from σ orbitals, given by the blue spectrum, have been subtracted from the original spectrum. The case of C6 H6 on Cu(110) shown in Fig. 1.20a reflects a relatively weak chemisorption bond reflected by the minor modification of the molecular benzene orbitals. This is in contrast to the case of C6 H6 on Ni(100) shown in Fig. 1.20b, where both XNLD and RIXS spectra directly reveal the distortion of the molecular orbitals upon chemisorption through changes of the spectral features in position and intensity [202].

32

1 Introduction and Overview

1.4.3 X-Ray Spectro-Microscopy The combination of x-ray absorption spectroscopy and microscopy is utilized in x-ray spectro-microscopy which yields element, bonding, and spin specific images of nanoscale structures [11, 204–206]. X-ray imaging techniques are based on either real space images or the inversion of reciprocal space diffraction images. As an example of the novel utilization of x-rays we briefly discuss their impact on modern magnetism research. Originally, the determination of magnetic structure was the domain of neutron scattering which first revealed the atomic spin structures of bulk single crystals. The progress of x-ray research may be put into perspective by the 1994 press release by the Nobel committee awarding the Prize in Physics to B. N. Brockhouse and C. G. Shull for the development of neutron techniques, stating: “Neutrons are small magnets ... (that) can be used to study the relative orientations of the small atomic magnets. (In contrast)...the x-ray method has been powerless, and in this field of application neutron diffraction has since assumed an entirely dominant position.” Ironically, at the time of the press release, the importance of x-rays for modern magnetism research had already been demonstrated by the discovery of x-ray magnetic dichroism effects in the late 1980s [174, 176] and their use for the study of magnetic nanostructures in the early 1990s [207]. By now, the use of x-rays has taken over from neutrons because of their larger spin dependent cross sections [34]. As an example we show in Fig. 1.21 images of magnetic domains recorded with photoemission electron microscopy (PEEM) [204, 208]). Figure 1.21a shows images of the ferromagnetic domains (“bits”) written on a CoPtCr coated disk in a computer hard drive [207]. The images were obtained with circularly polarized soft x-rays tuned to the Co L3 resonance. In (b) we show the naturally occuring antiferromagnetic domains in an epitaxially grown LaFeO3 film, recorded with linearly polarized xrays tuned to the Fe L2 resonance [209]. In (c) and (d) the correlation between the ferromagnetic domains in a thin Co film and the antiferromagnetic domains in a NiO single crystal underneath is demonstrated. In the figure the original assignment of the orientation of the antiferromagnetic axis [210, 211] has been changed by 90◦ based on more recent work [212–214]. The functioning of a magnetic memory cell is illustrated in Fig. 1.22 by time dependent nanoscale images recorded with scanning transmission x-ray microscopy (STXM). The experimental geometry is schematically shown in Fig. 1.22a [215], and the images recorded at the Advanced Light Source (ALS) in Berkeley in (b) [216]. They reveal the remarkable switching of the magnetic orientation of the blue layer in (a), whose direction represents either a 0 or 1 bit. It evolves by motion of a vortex core through the bit instead of a simple rotation of the entire magnetic bit. Not shown is that a current pulse sent in the opposite direction switches the magnetic bit back into the original direction [216, 217]. The images reveal the extraordinary capabilities of modern storage rings which, in contrast to XFELs, provide high intensity stability required for state-of-the-art measurements of small signals. Elemental specificity is used to pick out the signal of

1.4 The Scientific Power of X-Rays

33

X-ray images of magnetic domains ferromagnetic CoPtCr

(a)

antiferromagnetic LaFeO3

(b)

50 m

11 m

Ferromagnetic domains in Co coupled to antiferromagnetic NiO

(c)

(d)

2 m Fig. 1.21 PEEM images of magnetic domains recorded with polarized x-rays. a Images of magnetic bits in a CoPtCr magnetic recording medium recorded with circularly polarized x-rays tuned to the Co L3 resonance [207]. The arrows indicate the ferromagnetic alignment directions. b Images of antiferromagnetic domains in LaFeO3 recorded with horizontal linear polarization alignment and the photon energy tuned to the Fe L2 resonance [209]. The domains are only visible when there is an unequal projection of the polarization axis onto the two antiferromagnetic axes indicated by double arrows. c Ferromagnetic domains in a Co film following the antiferromagnetic domains of the NiO single crystal template underneath, shown in (d). The assignment of the orientation of the antiferromagnetic domains [210, 211] has been changed by 90◦ based on later work [212, 213]. The ferromagnetic images were taken with circular polarized x-rays tuned to the Co L3 resonance, the antiferromagnetic images of the same region utilized linear polarized x-rays tuned to the Ni L2 resonance

34

1 Introduction and Overview

Spin torque switching of magnetic nanodevice focused circ. pol. x-rays

(a)

Cu lead bipolar pulse generator

switching magnetic layer fixed magnetic layer

fast detector

t =0

(b)

+ current pulse

t =4ns

no current

100 nm

1.8 ns

2.0 ns

2.4 ns

Fig. 1.22 STXM images of the mechanism underlying the spin torque switching of a magnetic layer Co layer (blue) in a memory device. A schematic of the experimental arrangement is shown in a [215] and time-dependent magnetic images are shown in b [216]

a 2 -nm-thick Co layer in an oval 180×110 nm pillar buried under a total of 250-nmthick metal (mostly Cu). Magnetic contrast is achieved by use of circularly polarized x-rays, spatial resolution of 30 nm through x-ray zone-plate focusing, and temporal resolution through the 70 ps long pulses emitted by the x-ray storage ring. The x-ray pulses are synchronized with ultrafast electronics with 30 ps rise time.

1.4 The Scientific Power of X-Rays

Seeing the Invisible

35

Where are the atoms?

C O

Where are the electrons?

C N

Where are the spins?

Fig. 1.23 Illustration why x-rays are so useful. Top left: X-rays allow imaging the invisible, illustrated by a medical application and by an image showing the bit structure of a magnetic recording disk. Top right: X-Ray crystallography reveals where the atoms are in crystals, macromolecules, or chemisorbed molecules. Bottom left: X-ray absorption and emission (shown on left) spectroscopy can reveal the bonding orbitals on individual atoms and angle resolved photoemission spectroscopy reveals the electron states at the Fermi surface of a high temperature superconductor (right). Bottom right: Magnetic linear and circular dichroism studies reveal the spin structure in ferromagnetically coupled multilayers (left) and near ferromagnetic/antiferromagnetic interfaces (right)

1.4.4 Summary of Key Capabilities of Synchrotron Radiation We may summarize the capabilities of synchrotron radiation x-rays for the study of properties of matter during the first one hundred years as shown in Fig. 1.23. During the twentieth century, x-ray science primarily concentrated on the investigation of the static or equilibrium properties of matter. The first well-known application of x-rays shown in Fig. 1.23 is a result of their penetrating nature, allowing one to see below the surface of objects, utilized in medical x-ray imaging. This capability became a world wide sensation immediately after Röntgen’s first experiments, and unlike many other scientific discoveries, its importance was immediately apparent even to the lay person and played a role in Röntgen receiving the Nobel Prize only six years after his discovery. The second well-known use of x-rays is a result of the wavelength being of the same spatial dimension as the atomic building blocks of matter. This has made x-ray diffraction the most important technique for the unraveling of the atomic structure of matter, ranging from solids to organic matter. As an example, by 2018 of the about 145 thousand solved macromolecular structures entered into the worldwide protein data bank about 130 thousand were solved by use of x-ray diffraction [218] with the overwhelming majority using synchrotron radiation. While the first two applications utilize “hard” x-rays with energies of 10 keV or higher, “soft” x-rays with energies in the 100–1000 eV range also have important

36

1 Introduction and Overview

applications. Soft x-rays are utilized in photoemission, absorption, and emission spectroscopy due to their sensitivity to the electronic and magnetic structure associated with the atomic valence electrons that form the bonding “glue” between atoms [29, 34, 37]. The valence electrons are either probed directly by photoemission or indirectly through resonant transitions between core and valence electrons. In contrast to hard x-rays, the use of soft x-rays was long impeded by their absorption in air and carbon contamination of x-ray optics, leading to the existence of a soft x-ray spectral gap [219, 220]. This was overcome only around 1980 by use of synchrotron sources in conjunction with ultra-high vacuum optics and sample environments [220, 221].

1.5 Science with XFELs We conclude this chapter by highlighting key new capabilities of XFELs in different areas of science. We note that, today, scientific teams typically utilize all existing XFEL facilities (see Fig. 1.13) where “beamtime” can be secured through proposals. The examples illustrated below are therefore a testament of the spirit of international scientific collaborations.

1.5.1 Snapshots of the Atomic Structure of Matter: Probe Before Destruction As first suggested by Neutze et al. [222], the combined high intensity and short duration of XFEL pulses may be used to record snapshots of the atomic structure of a sample before it is destroyed by the high intensity pulse. The essence of the “probe before destruction” concept is based on the fact that the speed of light is about a million times faster than the speed of sound. In practice, a picture of the equilibrium atomic structure may therefore be captured by a femtosecond pulse before there is significant change of the atomic positions on the 1 picosecond timescale. The large deposited energy per pulse initially heats the electronic system on the femtosecond timescale before the energy is transferred to the lattice, leading to large thermally induced atomic motion that results in bond breaking, melting, and ultimately sample explosion [222]. This temporal evolution leads to the counterintuitive ability of XFEL pulses to actually avoid structural and crystallographic damage which may occur at longer exposure times required for synchrotron radiation [223], with the caveat that the sample needs to be replaced after each shot. This “probe before destruction” concept only applies, however, to the positions of the atomic cores (or nuclei) but not to the valence electronic structure since charge and spin densities can change on the timescale of order of 1–10 fs (see below).

1.5 Science with XFELs

37

Arrangement for typical XFEL diffraction studies large-angle CCD atomic structure

sample jet

lses

L pu

XFE

conv. laser pulses

interaction region

small angle CCD nanoscale structure

Fig. 1.24 Typical experimental arrangement for XFEL diffraction experiments, adapted from Chapman et al. [226]. A jet-based sample delivery system provides either a continuous-stream or pulsed beam. A sample response may be initiated by a short pulse conventional laser that also defines a time-zero reference for time-resolved XFEL diffraction patterns. A position sensitive CCD detector records the wide angle diffraction pattern, while the small angle pattern transmitted through a central opening is recorded with a second CCD detector

The requirement of shot-to-shot sample replacement led to the development of new sample delivery systems [224, 225], which are incorporated into a typical experimental arrangement for XFEL diffraction studies, as schematically illustrated in Fig. 1.24. Various kinds of samples can be delivered by a liquid jet as shown in Fig. 1.24, such as macromolecules and small protein crystals surrounded by water or different liquids, while atomic clusters may be delivered by gas dynamic virtual nozzles [224, 225]. For solid samples, pulse-synchronized sample rastering techniques may be used [148, 227] since the required motion is quite small due to the focused micron-size XFEL beam. Using the experimental geometry in Fig. 1.24, Chapman et al. [226] in 2011 successfully demonstrated the “probe before destruction” concept with atomic resolution. Instead of single protein molecules envisioned by Neutze et al. [222], they used small protein crystals which diffract significantly stronger. The diffraction patterns could also be directly compared to those recorded with synchrotron radiation. Another experiment in the same year by Seibert et al. [228] probed the shape and internal structure of single viruses on the nanometer length scale. These experiments established XFELs as new powerful tools for structure determinations of macromolecules in their natural room temperature environment, based on the counterintuitive trick to beat radiation damage by use of ultra-intense and ultra-fast x-ray pulses. The required size of protein crystals, whose growth is often an impediment, may also be reduced with the ultimate goal of single molecule imaging [222]. The XFEL-based development of protein crystallography has been reviewed by Brändén and Neutze [229] and Barends et al. [230].

38

1 Introduction and Overview

One area, where XFEL studies have already had a significant impact, is the understanding of arguably the most important chemical process on Earth, oxygenic photosynthesis. It is based on the action of an enzyme known as photosystem II, that uses solar energy to split water molecules, and in the process creating oxygen molecules plus electrons and hydrogen radicals needed to convert carbon dioxide into the organic fuel glucose, C6 H12 O6 , and different forms of hydrocarbons [231, 232]. About 2.5 billion years ago, this process resulted in our O2 -rich atmosphere, the ozone layer and eventually the evolution of multicellular lifeforms. Today, its detailed understanding is envisioned to provide a blueprint for future technologies based on using solar energy to satisfy our ever-increasing energy demands. Over the last twenty years, remarkable progress has been made in understanding this complex process in terms of the structure of the central oxygen evolving center, Mn4 CaO5 . As reviewed by Barber [231], an early picture was obtained with synchrotron radiation-based diffraction [233–235] and x-ray spectroscopic studies [236, 237], which has been increasingly refined through XFEL studies, as reviewed by Cox, Pantazis, and Lubitz [232] and Bergmann et al. [238]. Uncertainties associated with potential x-ray damage of synchrotron based studies were only overcome through XFEL studies, which revealed the atomic structure of the central Mn4 CaO5 cluster and its function during the so-called Kok cycle [239] shown in Fig. 1.25 [238]. Combined diffraction and x-ray emission studies revealed the key insertion of a new oxygen atom (open red circle) in the Mn cluster during the second flash 2F, followed by initial O–O bond formation coupled to Mn reduction after the third flash 3F as indicated in the figure [240].

1.5.2 Creation and Characterization of Transient States of Matter Another broad theme of XFEL applications is time dependent snapshots of fleeting atomic arrangements by diffractive imaging and of the associated electronic structure by x-ray absorption and emission spectroscopy. Below we briefly discuss the use of XFELs to detect and characterize transient states of matter in the form of chemical transition states in molecules and transient collective states in liquids.

1.5.2.1

Femtochemistry

The field of femtochemistry explores the very act underlying chemistry, the making and breaking of chemical bonds. It requires techniques that can image on the atomic length scale and the intrinsic femtosecond timescale over which the valence electron density (molecular orbitals) can change. The field was pioneered by Ahmed Zewail, who in 1999 was awarded the Nobel Prize in Chemistry “for his studies of the transition states of chemical reactions using femtosecond spectroscopy” [241]. The

1.5 Science with XFELs

39

The Kok cycle of photosynthesis 1F

S1 2 O 4 Mn 4

3 5

2 Ca 3

e–

S2

1 1 H2O

2F

e–,H + e–,H +

4F

S3 S0 new oxygen

H2O molecular + oxygen O2,H

(S4) e–,H +

3F

Fig. 1.25 Kok cycle of photosynthesis [239]. The overall structure of the protein (center) is surrounded by the four-step catalytic cycle (Kok cycle), revealed by light flashes 1F–4F. For each of the stable states S0 , S1 , S2 , and S3 , the x-ray diffraction structure of the catalytic Mn4 CaO5 cluster (O in red, Mn in purple, and Ca in green) obtained from XFEL measurements is also shown. Figure adapted from Bergmann et al. [238]

early work was based on combining the femtosecond time resolution of laser pulses with the atomic length scale resolution of electron diffraction. XFELs naturally have both, and this ability has given rise to the new field of ultrafast x-ray science. It has been argued that this has led to an ultrafast x-ray spectroscopic revolution in chemical dynamics [242]. The often quoted quest of femtochemistry is a molecular movie with femtosecond frames and atomic resolution that shows the breaking or formation of chemical bonds. While this goal will most likely be reached for simple molecules, the real problem of practical importance is much more complicated. It involves the understanding of catalytic pathways in multi-component systems that can inspire the development of new energy-efficient technologies. In practice, many chemical transformations are catalytically enhanced and proceed in multi-component systems from equilibrium structures and states, defined by energy wells in the potential energy surface, through transition states that exist at energy barriers along a reaction path. These transition states are hard to capture or observe not only because of their short lifetimes but because they may involve only tiny fractions of the equilibrium atomic populations. We are only beginning to tackle some simple problems in the vast area of chemical and

40

1 Introduction and Overview

biological activity. This is facilitated by the elemental and bonding state specificity of resonantly tuned x-rays. In practice, the observation of transition states also requires that the process is triggered on an ultrafast timescale so that its evolution can then be followed in time. This has naturally led to pump-probe spectroscopy studies of photochemical reactions triggered by optical laser pulses down to tens of femtoseconds durations. Optical pulses are also used in XFEL studies as a trigger, followed by femtosecond x-ray probe pulses, which can determine either the change in atomic positions by diffraction or bonding by spectroscopy as reviewed by Chergui and Collet [243] and Wernet [244]. By now, there are many examples of utilizing the combined capability of triggering a chemical reaction by an optical pump pulse and probing it with an XFEL pulse tuned to an atom-specific absorption resonance. By this method the bond breaking of carbon monoxide (CO), catalyzed by a ruthenium (Ru) surface, has been observed, followed by the creation of carbon dioxide (CO2 ) [165, 166]. Other studies have used a combination of x-ray absorption and resonant inelastic x-ray scattering (RIXS) to map out the detailed photochemical reaction in transition metal complexes such as Fe(CO)5 and [FeIII (CN)6 ]3− as reviewed by Jay, Kunnus, Wernet, and Gaffney [245].

1.5.2.2

Transient Density Fluctuations

It is well known that nanoscale ordering phenomena associated with charge, spin, and orbital order exist in correlated materials [246]. In many cases these states have sufficiently long lifetimes to be imaged by diffraction or real space microscopy. It has recently been observed, however, that charge order in La2−x Bax CuO4 exhibits fluctuations on picosecond timescales accessible with XFELs [247]. Since picosecond temporal fluctuations correspond to meV energy changes that are difficult to resolve by inelastic x-ray scattering in the energy domain, time-dependent x-ray scattering measurements with XFELs appear to provide important new capabilities. Similar to correlated materials, liquids where small molecular units interact loosely through hydrogen bonds are also expected to exhibit nanoscale order associated with changes in orientational order, conformation, and atomic density within the first few coordination shells of the individual molecules. Such systems may appear globally homogeneous, and the local structural motives may be difficult to detect owing to their dynamic fluctuations. Judging from Fig. 1.12, the laws of inertia would seem to allow fluctuation of molecular groups on sub-nanosecond timescales. Such locally different structural motives are today considered as an explanation of the unusual thermodynamic properties of liquid water. Water is the most important liquid for our existence and is the only common substance that appears under everyday conditions in all three states of aggregation—gas, liquid, and solid. It is essential for many energy-related chemical transformations and environmental processes. From a scientific point of view, water is a liquid with strange thermodynamic properties; its maximum density occurs at 4◦ C and increases upon melting (ice floats

1.5 Science with XFELs

41 The structure of liquid water

(a)

?

?

(b)

(c) Detector

droplet dispenser

(d) Detector

X-ray pulses X-ray pulses

X-ray pulse

IR heat pulse

IR heat pulses

Fig. 1.26 a Schematic phase diagram of water connecting the liquid, supercooled, and glassy states labeled low density amorphous (LDA) and high density amorphous (HDA). The gray-colored area depicts the “no-man’s land” which is conjectured to contain an illusive critical point, indicated in blue. The left inset shows two proposed fluctuating phases, separated by the dashed red Widom line, that consist of different mixtures of LDA and HDA, based on the latest XFEL diffraction studies [251, 252]. b, c Schematic diagrams of preparing supercooled water in the no-man’s land region. b A continuous series of ∼ 14 µm water droplets cooled through evaporation as a function of distance between the dispenser and the measurement point. c Heating of the supercooled water droplets with infrared pulses before the x-ray pulses. d Ultrafast heating of amorphous ice samples with infrared pulses. Figure courtesy of Anders Nilsson [252]

on top), its viscosity decreases under pressure, and it has a high surface tension, to name a few. The structure of water was already studied by x-ray diffraction in the 1930s, and in 1933 Bernal and Fowler [248] deduced an essentially tetrahedral structure in which each water molecule has four nearest-neighbor molecules. They concluded that modifications of the diffraction patterns with temperature from below 4◦ C (ice) to 340◦ C indicated subtle structural changes that evolved continuously into each other. Today, the complexity of possible networks of H2 O molecules is evidenced by the fact that there are 17 experimentally confirmed polymorphs of ice, with snowflakes corresponding to one of them [249]. Over the years, the liquid phases of water have attracted particular attention with experimental neutron and x-ray studies complemented by molecular dynamics computations as reviewed by Nilsson and Pettersson [250]. The pressure-temperature phase diagram emerging from these studies is shown in Fig. 1.26a.

42

1 Introduction and Overview

Without going into details, the key to a uniform structural and thermodynamic picture appears to be the existence of a critical point, indicated in blue, with the extension of a so-called Widom line,18 shown in dashed red, to a temperature of 228◦ K at 1 bar. Remarkably, water in the form of small droplets can be super cooled to a temperature close to this value without crystallization. Over the last ten years or so, much effort has gone into exploring the so-called no-man’s land indicated in gray in Fig. 1.26a, which is believed to contain the indicated critical point (blue circle). The combination of diffraction experiments with sub-100 fs pulses, where each pulse yields a diffraction pattern that can be separately analyzed, together with novel sample preparation techniques, illustrated in Fig. 1.26b–d has allowed access to the gray region for the first time, as reviewed by Nilsson and Perakis [251, 252]. These studies have proposed two fluctuating phases, shown in the inset in Fig. 1.26a, that consist of different fluctuating mixtures related to two pure macroscopic liquid phases of the two glassy states labeled low density amorphous (LDA) and high density amorphous (HDA).

1.5.3 Creation and Probing High Energy Density Matter Remarkably, x-rays interact more strongly with atomic core than valence electrons, as illustrated for Ne atoms in Fig. 1.27. The figure shows that at x-ray energies that exceed the binding energies of the innermost 1s electrons, the interaction cross section with different atomic shells decreases from the inside out. This allows the creation of “hollow atoms” with x-rays, as beautifully demonstrated in early XFEL experiments [255, 256]. In this context it is important that the cross sections shown in Fig. 1.27 are single photon absorption or photoemission cross sections. Once an electron has been excited from a given core shell, the excitation probability of additional electrons from the same core shell is exceedingly small [256]. The stripping of electrons from atomic shells also occurs in solids. After a femtosecond pulse excitation, a solid is turned into an electronically hot state. However, atomic motion and bond breaking will only occur on picosecond or longer timescales because of the greater mass of atoms or ions than electrons, ultimately ending in the explosion of the sample. The properties of such created hot plasma-like states are of interest in the field termed high energy density (HED) science [257]. This relatively new subfield of physics has strong connections with plasma physics, nuclear physics, and astrophysics [258, 259]. HED matter may be created with multi-terawatt lasers and more recently with XFELs. The short XFEL pulses can both create and probe the HED state, and their complete penetration provides true “bulk” information. This is a significant advantage over optical pulses, which after creation of the HED state cannot penetrate it. For the combined reasons of being ultrashort and penetrating, XFEL pulses have become powerful new tools for HED science. 18

The Widom line, named in honor of Benjamin Widom, defines a line in the pressure-temperature plane where the correlation length has its maximum.

1.5 Science with XFELs

43

10 2 10

vacuum level

0

2p

- 21.6

2s

- 48.5

Cross section (Mb)

Energy (eV)

1

2

Cross section (Mb)

(b) Ne cross sections

(a) Ne atomic shells

1s Rydberg fine structure

1.5 1 0.5 0 865

867 869 Photon energy (eV)

10 -1

1s

10 -2

2s

871

2p

10 -3 1s

- 870.2

10 -4 0

200

400

800 1000 1200 600 Photon Energy (eV)

1400

Fig. 1.27 Photon excitation cross for different shells of Ne atoms. The atomic shell structure and binding energies are shown in (a), and the respective ionization cross sections are shown in (b) [253]. The inset shows the fine structure due to Rydberg resonances just below the 1s (K-shell) ionization threshold at 870.2 eV [254]

An early example that directly reveals the evolution of the simple metal Al from room temperature to the HED regime is shown in Fig. 1.28. Al has two core absorption edges, an L-edge (2 p excitation) at 73 eV and a K-edge (1s excitation) at 1560 eV. The shown work [260, 261] utilized ω =93 eV XUV radiation from the FLASH facility.19 Figure 1.28a shows the change in transmission through an Al film as a function of fluence for incident 15 fs pulses of ω = 93 eV or 20 eV above the lowest energy L2,3 absorption threshold. With increasing fluence, the sample is seen to become nearly transparent. This is caused by increasingly stripping single electrons from the 2 p core shell until at the highest fluence all illuminated atoms have a 2 p core hole. Then further absorption becomes prohibited since the excitation probability of a second electron out of the 2 p core shell of the same atom is negligibly small as shown by early XFEL experiments [255, 256].20 Also, the 2 p core shell excitation dominates over direct excitation of valence electrons (see Fig. 1.27). The changes

19

Later work also explored the K-edge response [262] but will not be discussed here. The excitation of more than one-electron from a core shell of atoms only occurs after all valence electrons are removed, as first observed in He-like Ne8+ [256].

20

44

1 Introduction and Overview (b) L-shell emission spectra

(a) X-ray transmission of Al film Al film, 53nm h = 92eV 15 fs pulses

0.6

0.4 0.3 1

0.2 1 10 100 Incident fluence 15 fs pulses (J/cm²) 2

Al IV emission lines

10

0.5

0.1 0.1

TW/cm2 = 5.1×103

3

1000

Emission intensity

0.7

Electronic temperature Te (eV)

Rel. film transmission

0.8

EF

1.5×10

kBT= 1.1 eV

2

0.9 eV

43

4

10 10 10 10 Incident intensity (TW/cm² = mJ/cm² /fs) 0.1 1 10 Photons per atom in film

9.3×102

100

5.7

55

0.4 eV

60

65 70 75 Emission energy (eV)

80

Fig. 1.28 Response of Al metal as a function of incident intensity of incident 93 eV pulses, about 20 eV above the lowest energy L2,3 core shell. a Change in transmission [260] as a function of fluence of 15 fs pulses. The highest fluence for the used 15 fs pulses of 200 J/cm2 corresponds to an intensity of 13 J/cm2 /fs (1.3 × 104 TW/cm2 ). The circles represent the experimental data points, and the solid line is a theoretical prediction. The dashed curve refers to the electron temperature in eV (right scale) of the valence electrons at the end of the 15 fs FEL pulse, but before the L-shell holes are filled at about 40 fs after creation. b Change of the emission spectrum as a function of intensity (fluence/pulse length) for 35 fs excitation pulses, revealing the transition from a transiently hot solid to the HED regime [261]. The blue curves are simulations and shown in green is the position of the Fermi level E F where changes in electronic temperature manifest themselves

in transmission reflect the saturation of the single 2 p core electron excitations per atom.21 The complementary emission spectrum shown in Fig. 1.28b shows that intensities below about 1.5 × 102 TW/cm2 , causing an electronic temperature in the sample of up to kB T  1 eV, modify the band occupation in Al near the Fermi level, E F , reflected by the change in slope of the Fermi-Dirac distribution around E F . At the highest incident intensities, the emission spectrum is dominated by intense plasma line emission that obscures any possible survival of valence band structure.

1.5.4 Non-linear X-Ray Interactions with Matter The field of non-linear (NL) optics was born with the advent of the laser, and its beginning is often associated with the discovery of second harmonic generation in 1961 by Franken et al. [264]. It is typically defined classically as the case where the 21

Since the created core holes, which create Auger electrons, decay on a longer time (30 fs [263]) than the pulse length, this process is not seen in the shown transmission measurement.

1.5 Science with XFELs

45

material response not longer depends linearly on the action of the electric field of light [265]. In the language of x-ray science and quantum optics, used in this book, NL optical processes may occur by the presence of more than a single photon within the atomic coherence volume of the atomic building blocks of matter. There is a remarkable exception to the above definition of NL x-ray science, based on the requirement of more than a single incident photon at a time. In 1971, Eisenberger and McCall [266] conducted an experiment where radiation from a 17 keV Mo x-ray tube source was scattered by a beryllium crystal. After about two weeks of accumulation of coincidence events in the scattered x-rays, they observed the splitting of single incident photons into two entangled photons, referred to as spontaneous parametric down conversion. This tour de force experiment, conducted about 40 years before the advent of XFELs, may rightfully be considered the first NL x-ray experiment. The key to understanding NL interactions of XFEL x-rays with matter is the photon degeneracy parameter whose increase with the advent of optical lasers and XFELs is shown in Fig. 1.3. It exceeds unity only for such sources. This means that all synchrotron radiation experiments are basically determined by interaction processes with matter that occur one-photon-at-a-time since the probability that two or more photons are simultaneously present is negligibly small. XFELs therefore offer the opportunity for the first time to study processes that are driven by two or more photons. This case was first treated quantum mechanically by Maria Göppert-Mayer in 1931 [267], who derived the expressions for two-photon absorption and photoemission and stimulated Rayleigh/Raman scattering, long before such processes could be observed.22 Over the last 15 years or so, different non-linear x-ray interactions with matter have been observed and non-linearity thresholds for XFEL radiation have been discussed in [268–272]. The observation of x-ray transparency in non-resonant absorption by Nagler et al. in 2009, discussed above, represents the first XFEL experiment revealing a NL response [260, 261]. It was followed by the demonstration of the creation of multi-electron excitations and the creation of hollow Ne atoms in 2010 [255, 256]. These studies were later extended with atomic and molecular [273, 274] and solid samples [259, 262, 275–277]. Another milestone is the study by Rohringer et al. in 2012 [278], demonstrating amplified spontaneous emission in Ne gas (see below). It was followed by the demonstration of stimulated x-ray emission [271, 278–282] and stimulated resonant inelastic scattering [283–286]. Other NL studies involved non-linear mixing of optical light and x-rays [287], second harmonic generation [288], two-photon absorption [272, 289, 290], four wave mixing [291], and anomalous Compton scattering [292]. Below we give two brief examples of XFEL induced non-linear processes.

22

I would like to thank Shaul Mukamel for pointing this out to me.

46

1.5.4.1

1 Introduction and Overview

Amplified Spontaneous Emission

A unique capability of XFELs is their ability to create a population inversion in atoms, similar to the first step in an optical laser. This requires an intense short pulse, or a spike within a pulse, whose intensity is sufficient to excite a significant fraction of atoms in a sample before they can decay again. For x-rays, this process involves kicking out core electrons from atoms, resulting in an excited atomic state that has a core hole. The process has to happen on a timescale shorter than the core hole decay time (a few femtoseconds). Following such a pump pulse, x-rays will first be spontaneously emitted during the decays of electrons from outer shells into the core hole. In his original introduction of spontaneous and stimulated emission, Einstein already considered the emission directions of spontaneous and stimulated radiation based on arguments of momentum conservation [60]. Spontaneous emission is in random directions, while stimulated emission is directional, with the photon driving the stimulated decay cloning itself in every aspect. By using a gas cell shaped as a cylinder, this cloning process will preferentially occur in the long direction of the cylinder, and consecutive stimulation events will lead to a directional photon avalanche, as in a laser cavity. The enhancement is referred to as amplified spontaneous emission (ASE) [293]. The x-ray ASE effect, first demonstrated by Rohringer et al. [278], utilized the geometry illustrated in Fig. 1.29. XFEL SASE pulses of 960 ± 4 eV photon energy,  50 fs duration, and an intensity up to 2 × 1017 W/cm2 (200 J/fs/cm2 ) were used to induce a population inversion of the Ne Kα transition (849 eV). The Ne gas was contained in an elongated cell along the beam direction which allowed the spontaneous Kα x-rays created in the front part of the cell to be amplified through stimulated decays along the incident beam direction. The 960 eV pump and 849 eV stimulated beams were then separated by a grating spectrometer of about 2 eV resolution and imaged by a CCD detector. A strong increase of about four orders of magnitude of the spontaneously emitted Kα radiation by the ASE process was observed upon doubling the incident pulse energy from 0.12 to 0.24 mJ.

1.5.4.2

Stimulated Forward Scattering and Loss of Diffraction

Rather than letting the stimulation process be triggered by uncontrolled spontaneous decays as in ASE, one may utilize a high intensity XFEL pulse whose temporal coherence length is longer than the core hole decay time. Such a pulse, created, e.g. by sending a SASE pulse through a monochromator, can then control both the excitation as well as the core hole decay process. The pulse necessarily has a narrow energy width and may be tuned to an atomic absorption resonance. When such a pulse is sent through a sample in the form of a thin film, with its energy tuned to an atomic resonance, the random spontaneous scattering observed at synchrotron sources will be replaced with increasing XFEL intensities by stimulated emission into the forward direction of the incident photons.

1.5 Science with XFELs

(a)

47

960eV ionization limit 870.2 eV 2p

1s

(b)

stimulated transition K 849eV core state

Ne gas cell (500 torr)

ASE 1.4 cm

XFEL pump 50 fs, 960 eV SASE pulse,

Fig. 1.29 a Energy levels and transitions of interest in Ne atoms. b Experimental scheme used to demonstrate amplified spontaneous emission (ASE) in a Ne gas cell [278]. The coherent spikes in the incident SASE pulse of about 50 fs and 960 eV nominal energy were of sufficient intensity to create K-shell holes in Ne atoms, overcoming their natural 2.4 fs decay lifetime. The spontaneously emitted Kα line of 849 eV by atoms at the front end of the cell is then amplified along the cell in the ASE process and separated from the incident 960 eV photons by a grating spectrometer to become visible on the CCD detector

If the thin film sample now contains nanoscale domains in either composition, thickness, bond orientation, or magnetization directions, the spontaneous diffraction pattern, observed at finite momentum transfer with a synchrotron source, will vanish with increasing XFEL intensity since all photons are scattered through stimulated decays in the forward direction, only. At sufficiently high intensity, absorption and stimulated emission will become the same and the film becomes transparent. Such self-induced transparency was first observed in the optical regime in 1967 [294, 295]. In the x-ray region, this effect was predicted theoretically by Stöhr and Scherz [296] and the disappearance of the diffraction pattern of magnetic domains in Co/Pd multilayer sample was demonstrated experimentally in 2016 by Wu et al. [297]. They used 50 fs SASE XFEL pulses, monochromatized to a bandwidth of 0.2 eV, and tuned to the Co L3 resonance (see inset of Fig. 1.17). In Fig. 1.30a we show the arrangement used in a later experiment by Chen et al. [298] with SASE XFEL pulses as short as 5 fs. They simultaneously monitored the interplay between the centrally transmitted and out-of-beam diffracted intensity by a similar Co/Pd film with nanoscale magnetic stripe domains.

48

1 Introduction and Overview

(a) Schematic experimental geometry YAG screen picture frame samples

Intensity

grating

Co/Pd

0.7 mm SiN 8 mm

Si frame

entrance slit

770 780 790 Photon energy (eV)

spectrometer 25 fs, 778 eV SASE pulses

beam splitter

CCD detector

focusing mirrors

E

Relative diffraction contrast (%)

spectrometer difference signal 100 80 60 40 20 0 10

100 80 60 40 20 0

Resonant transmission (%)

(b) Out-of-beam diffraction and central beam t ransmission

relative CCD diffraction signal 3 10 10 2 2 Incident intensity (mJ/cm /fs)

Fig. 1.30 Experimental geometry for simultaneous pulse-by-pulse measurements of the transmitted and diffracted response of a Co/Pd thin film [298]. Incident SASE pulses of ∼ 25 fs length are focused onto the sample plane and split by the sharp edge of a mirror, with one half propagating through a Co/Pd/SiN sample in a picture frame and the other through a pure SiN reference film for normalization purposes. The horizontal magnetic stripe domains in Co/Pd produce strong first and third order Bragg diffraction peaks on a pnCCD detector. The spatially separated undiffracted beams are allowed to propagate into a downstream spectrometer. A grating disperses the two offset beams onto a yttrium aluminum garnet (YAG) fluorescence screen, yielding separate single-shot sample and reference spectra around the Co L3 resonance. The shown first and third order diffraction images on the CCD detector and spectra on the YAG screen are real data averaged over many pulses

1.5 Science with XFELs

49

Figure 1.30b shows the statistical average of the first order diffracted intensity (red curve), corresponding to the small intensity spots on the CCD separated by 8 mm in (a), and the normalized sample transmission signal (blue) corresponding of the peak value of the normalized Co L3 absorption signal, represented in (a) by the size of the central dip in the YAG screen intensity. The two signals show complementary behavior with increasing incident intensity, reflecting the increase in sample transmission at the cost of the loss of the out-of-beam diffracted intensity. At the highest intensity the sample has become transparent and stimulated forward scattering has replaced conventional spontaneous out-of-beam diffraction.

1.5.4.3

A Historical Perspective

It is interesting to put the last experiment into perspective of the historical evolution of stronger sources of electromagnetic radiation. As discussed in Sect. 1.2.5, NMR was facilitated by the development of stronger RADAR microwave sources in the 1940s, which in the late 1950s was followed by the advent of the maser/laser, and another 50 years later by the development of XFELs. The key breakthroughs with these sources were the creation of intense coherent radiation with degeneracy parameters far exceeding unity. In each case, the coherent control of transitions between two quantum states became possible for the first time. Coherent control means that one may picture the concerted action of many photons, generated within a source or coherence volume of order λ3 , as a strong classical wave that can cause both up and down transitions in a two-level system. The cramming of photons into the coherence volume becomes increasing difficult with decreasing λ, accounting for the timeline of the historical evolution. Another way of putting this into perspective is shown in Fig. 1.31, revealing the coherent control of quantum states with increasing energy separation, from μeV in NMR to eV for lasers to keV for XFELs. The figure also illustrates the electromagnetic aspect of radiation. In NMR one utilizes the magnetic component of a strong coherent microwave field to drive transitions between nuclear states split by a static magnetic field as illustrated in Fig. 1.31a. The process involves a nuclear spin flip and is described by the famous Bloch equations [299]. For the laser case, shown in Fig. 1.31b, the two levels are electronic valence states, with the transition driven by the electric component of the optical field. At optical frequencies, the magnetic field component that acts on spins becomes unimportant since the electronic spins can no longer follow the rapid oscillatory motion, and the process is described by the optical Bloch equations [15, 300]. With the advent of XFELs, multi-photon or strong field effects allow the control of atomic core-to-valence transitions in the keV range as shown in Fig. 1.31c.

50

1 Introduction and Overview NMR nuclear states

Optical Laser X-Ray Laser valence states core-valence states

sz 1 eV

1 eV

1 keV

Field driven transitions between quantum states m m

H

nm

E

E

Fig. 1.31 Historical milestones due to the development of improved sources of electromagnetic radiation. The development of radar in the 1940s enabled control of spin transitions through the H-field in nuclear magnetic resonance. This development was extended through the invention of the maser and laser into the eV range, corresponding to E-field driven transitions between electronic valence states. The advent of XFELs has allowed the control of E-field driven core-to-valence transitions in atoms in the keV range

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Part I

Production of X-Rays and Their Description

Chapter 2

Production of X-Rays: From Virtual to Real Photons

2.1 Introduction In this chapter we examine the fundamentals of accelerator-based x-ray sources and how the quality of the emitted radiation by many electrons contained in an electron bunch can be described through the equivalent concepts of brightness1 and coherence of a source. In addition of the conventional description of the generation of electromagnetic (EM) radiation, given in books [1–5] and reader-friendly reviews [6–9], we also introduce a different treatment. It is based on the conversion of a virtual photon cloud attached to a charge into real photons when the charge experiences a change in its velocity. This method, called the Weizsäcker-Williams (WW) formalism [10, 11], describes how the Coulomb fields attached to a relativistic electron bunch are converted into freely propagating electromagnetic radiation. The WW concept developed in the mid1930s foreshadowed the formal development of QED in the late 1940s. The central idea is that the Coulomb or “velocity” fields of a charge in steady motion [12] may be thought of as a cloud of virtual photons attached to and moving with the charge. The energy stored in the Coulomb field is equated to that of the virtual photon cloud which may be partly converted into true photons or “radiation” when the constant velocity of the charge is changed. This happens, for example, when electrons are slowed down by a target, resulting in bremsstrahlung radiation or accelerated by a magnetic field, leading to synchrotron radiation. The WW formalism conveniently allows an early introduction of the concept of virtual photons which are the carriers of the electromagnetic force and in QED constitute the quantum mechanical zero-point field. It also yields a simple geometrical picture of coherence based on the concept of a coherence cone, in which the EM fields of the individual electrons in a bunch add up coherently. The chapter also 1

The term “brightness” is typically used in the USA, while the equivalent term “brilliance” is used preferentially in Europe. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_2

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discusses the characteristics of synchrotron radiation emitted by bending magnets and undulators and compares them to the radiation properties of XFELs. The concepts used today to describe the EM fields associated with a charged particle that moves with an arbitrary velocity, including the effects of relativity, date back to around 1900 when Liénard [13] and independently Wiechert [14] derived a mathematical formulation now referred to as the Liénard–Wiechert field equation [2, 12, 15]. At the time, the electron had just been discovered by J. J. Thomson in 1897 and the difference between Lenard’s cathode rays (electrons) and Röntgen’s x-rays was known. The work of Liénard and Wiechert was extended in 1912 by Schott [16], who already conjectured the existence of what we now call synchrotron radiation. EM radiation is produced by accelerating charges and the higher the energy of the moving charge the higher the energy of the radiation that can be produced. In the late 1960s and early 1970s, the advantages of using electron accelerators for the production of EM radiation in the form of x-rays became clear [17]. In accelerator-based sources the kinetic energy of electrons is increased from tens of keV in conventional x-ray tubes to a few GeV, surpassing the relativistic limit set by the rest mass energy of the electron, m e c2 = 511 keV. The relativistic reduction of the Coulomb repulsion between the electrons allows the generation of electron bunches of high charge densities and the emission of radiation into a narrow forward cone when the electrons are radially accelerated in a magnetic field. Over time, various generations of synchrotron radiation (SR) sources have improved the quality of the emitted x-rays. The quality of a source is quantitatively described by its brightness, which cannot be improved but only spoiled by the manipulation of x-rays through optics. The fundamental importance of brightness and its direct link to the non-trivial concept of what we call “coherence” will be discussed in detail in Chap. 4. It will then become clear that “coherence” can be more accurately defined through increasing orders of perturbations within quantum electrodynamics. At this point it suffices to use the word “coherence” in the conventional sense defined through wave interference, which corresponds to first order coherence in QED [9]. Owing to the energy width of the emitted spectrum, SR sources are never intrinsically temporally coherent, also called transform limited, and can only be made so by use of a monochromator. The quality of a SR source is given by its average brightness, defined by arbitrarily assuming a spectral bandwidth of λ/λ = 0.1. Today’s SR sources are also not laterally coherent at higher photon energies. The goal of so-called ultimate storage rings is to achieve lateral coherence or diffraction limited performance over the photon energy range up to tens of keV [6, 8]. Figure 2.1 illustrates the limitations in lateral coherence of third generation storage rings relative to a completely coherent “ultimate” ring. The difficulty of achieving a diffraction limited source lies in the effect of the emitted radiation on the electron bunch. In storage rings, an equilibrium is established between energy and momentum transfer between the electron and photon systems, which limits the achievable size and angular emission cone of the electron bunch. This is overcome in a linear accelerator (linac), where the radiation losses are sufficiently small that it is possible to prepare an exquisitely small electron bunch before it is made to radiate in an undulator. This ability led to the development of x-ray free electron

Fig. 2.1 Illustration of the laterally coherent fraction as a function of photon energy, for a hypothetical diffraction limited ring, and that available at a characteristic third generation source and for an envisioned “ultimate storage ring” [3, 6, 8]

Laterally coherent fraction

2.2 Relativistic Concepts and Electron Bunch Compression 1.000 0.500

63

ultimate ring

0.100 0.050 third gen. ring 0.010 0.005 0.001 100

200

500

1000 2000

5000

10 4

Photon energy (eV)

lasers (XFELs). In the most common self-amplified spontaneous emission (SASE) XFELs, a well-prepared bunch of micrometer dimensions is sent through a long (tens of meters) undulator before it is discarded. Since each bunch is freshly prepared, the most brilliant radiation comes at an expense of a repetition rate (∼ 102 −105 Hz) which is much smaller than the 500 MHz rate of a storage ring. As a compromise between storage rings and XFELs, energy recovery linacs (ERLs) have been proposed [18, 19]. Such sources integrate a linac acceleration structure into a long straight section of a race track shaped magnet lattice resembling that of a storage ring. After an individual high-quality electron bunch is prepared in the linac section, it is allowed to circle once around the race track where it can serve many user stations. During its single cycle, the electron bunch maintains its properties since it requires thousands of cycles for a freshly prepared bunch to reach an equilibrium state. At the end of its cycle the bunch passes through the same long linac section and is decelerated to recover part of its energy before it is discarded in a beam dump. Energy recovery helps to reduce the operational costs of the facility. Energy recovery linacs have high repetition rates and therefore high average spectral brightness, but lower peak brightness than an XFEL.

2.2 Relativistic Concepts and Electron Bunch Compression Before we derive the fields associated with relativistic electrons in accelerators, it is useful to briefly review some key relativistic concepts. In the description of electron beams we use the well-known length and time transformations between inertial systems that move relative to each other with a constant velocity v [12, 20]. We define the rest frame of an object as the coordinate system in which it is at rest. The length measured in the rest frame is called the proper length, and the time measured by a clock in the rest frame is called the proper time. Following convention, we define the following relationships involving the ratio of v and the speed of light c,

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2 Production of X-Rays: From Virtual to Real Photons

v β≡ c

 1

γ ≡ 1 − β2

β≡

1−

1 1 1− . 2 γ 2γ 2

(2.1)

With these conventions the relativistic energy E e of the electron beam in the laboratory system, which is the sum of the electron kinetic energy Ekin = m e v 2 /2 and the rest energy of the electron m e c2 = 0.511 MeV, is given by2 γ =

Ee , or with Ee in [GeV] : m e c2

γ = 1.96 × 103 Ee

(2.2)

At a moderate accelerator energy of ten times the rest mass energy or 5 MeV, we have γ  10 and the particle travels already at v = 0.995 c. As the particle energy is further increased, the particle velocity remains relatively constant at about the speed of light. In a linear accelerator, the electron velocity thus remains nearly constant once its energy is well above the electron rest mass energy of 0.511 MeV. Since energies of tens of MeV are considered small for typical high energy physics accelerators, one may consider the electrons in uniform motion through most of their path.3 If we compare the proper length l0 in the rest frame to the length l along v, in a system moving with v relative to the rest frame, the lengths are related by the well-known Lorentz relationship l0 (2.3) l= . γ If l0 and l are measured perpendicular to v, we have v⊥ = βc = 0 and l0 = l. This shows that the length is always the largest in the rest frame, which is the well-known Lorentz contraction observed for a moving system. For an electron beam of 5 GeV we have γ = 104 and an electron bunch that appears to be l = 1 mm long in the laboratory frame, has a proper length l0 = 10 m in the rest frame of the electron bunch. This bunch has a temporal length of t = l/c  3 ps in the laboratory frame. The time behaves exactly opposite. If we measure a time t0 in the rest frame, the time t in the moving frame is (2.4) t = t0 γ . The proper time measured in the rest frame is therefore always the shortest, and we observe a time dilation for a moving system.

The electron mass m e in SI units is given by m e = 9.109 × 10−31 VAs3 m−2 = 9.109 × 10−31 kg. Further increases in energy originate from the increasing relativistic mass m ∗e of the particle ∗ m e = γ m e . In Einstein’s theory of relativity, there is a crucial difference between particles with and without mass: All massless particles must travel at the speed of light, whereas massive particles can never attain this ultimate speed. 2 3

2.3 The Fields of a Moving Charge: Liénard–Wiechert Equations

65

It is interesting and important that relativistic effects for the Coulomb forces between electrons exist both in the direction of motion and perpendicular to it. When two electrons are separated by a distance d0 in their rest frame along the direction of motion v relative to an observer, the distance d seen by the observer is strongly contracted but the force is unchanged, d =

d0 , γ

F = F 0 .

(2.5)

Thus through relativity one can produce for the observer an apparent small separation between the electrons without changing the electrostatic repulsion. When two electrons are separated by a distance d0 in their rest frame perpendicular to the direction of motion v relative to an observer, the distance d⊥ seen by the observer is unchanged but the force is reduced according to d⊥ = d0 ,

F⊥ =

F0 . γ

(2.6)

This remarkable result is due to the fact that in the perpendicular direction the force has both an “electric”, F⊥E , and “magnetic”, F⊥B component with opposite signs, and therefore the magnetic contribution weakens the Coulomb repulsion. Thus the bunch can also be compressed in the perpendicular direction. In practice, remarkable bunch compression results have been achieved at the SLAC National Accelerator Laboratory using its 3-km-long linear accelerator (linac) before it was converted for LCLS. The linac was capable of generating ultrarelativistic electron beams with energies up to 50 GeV. At an energy of 28 GeV, a Gaussian rms bunch length as short as σz = 21 µm in the laboratory frame was achieved while maintaining Ne = 1.5 × 1010 electrons in the bunch [21, 22]. Such bunches pass by a point in the laboratory in a time span 2σt = 140 × 10−15 s or 140 femtoseconds (fs) and therefore create a peak current of I  Ne e/2σt  1.5 × 104 A. Because the bunch can also be compressed in the perpendicular direction, high current densities can also be obtained. In fact, sub micron beam diameters have been achieved. Denoting the lateral cross section (area) as a, the current density is given by j = Ne e/(2σt a), and values of j  1016 A/m2 have been reached [23]. The manipulation of the shape and size of relativistic electron bunches has become a science in itself and so has their accurate characterization [24].

2.3 The Fields of a Moving Charge: Liénard–Wiechert Equations The calculation of the fields associated with a single charged particle in arbitrary motion, including the effects of relativity, was first accomplished by Alfred-Marie Liénard (1869–1958) in 1898 [13] and independently by Emil Wiechert (1861–

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2 Production of X-Rays: From Virtual to Real Photons

1928) in 1900 [14]. In their honor, the mathematical formulation of the fields is called the Liénard–Wiechert field equation. We shall not derive the Liénard–Wiechert expression here but will simply state it and use it. For a detailed discussion the reader is referred to other texts [2, 12, 15]. The book of Hofmann [2] provides a particularly nice derivation of the field equations and their application to different situations. The tricky part in the calculation of the fields is the treatment of the relativistic effects referred to as “retardation effects”. Retardation effects arise from the finite velocity of light. Let us for a moment assume that the velocity of light was infinite. We could then describe all fields in a “snapshot picture” that instantaneously correlates time and position. For example, the E-field measured at a given time could be directly expressed as a function of the position of the charged particle at the same time, as in electrostatics. However, since the speed of light is finite, there is a time difference between the emission and observation of the light. As illustrated in Fig. 2.2, it takes the light a finite time Δt = r ∗ /c to travel the distance r ∗ . We can then no longer simply write down an expression for the field E(t), measured at an instantaneous time t, in terms of the particle properties (e.g. position, velocity, and acceleration) at that time since the relevant particle properties that determine E(t) occurred at an earlier or retarded time t ∗ = t − r ∗ /c. In order to calculate the field at an observer at time t we must know the position r(t ∗ ) = r∗ , the velocity v(t ∗ ) = v∗ , and the acceleration a(t ∗ ) = a∗ of the charge q at the retarded time t ∗ . In our notation, the relation between t and t ∗ is given by the following two general equations [2]

origin

x0 = const.

x "retarded" position at t* = t - r*/ c

* r (t

)=

observation at time t

r*

n* v(t* )= v*

a(t* )= a*

“present” position at t Fig. 2.2 Illustration of the retarded time concept. A charge moves on an arbitrary trajectory. The fields at the observer at a time t are determined by the position r(t ∗ ) = r∗ , the velocity v(t ∗ ) = v∗ = dx/dt ∗ = −dr∗ /dt ∗ , and the acceleration a(t ∗ ) = a∗ at the retarded time t ∗ = t − r ∗/c. The observer is located at a fixed position x0 relative to the origin of the coordinate system

2.3 The Fields of a Moving Charge: Liénard–Wiechert Equations

t = t∗ +

r∗ , c

  n∗ · v∗ dt = 1 − dt ∗ c

67

(2.7)

These simple equations, however, may have very complicated solutions for certain particle motions! Liénard and Wiechert succeeded in deriving equations that express the electric and magnetic fields of a point charge in arbitrary motion. Below we give these equations using the notation of Fig. 2.2, where at the time of emission the unit vector n∗ = r∗/r ∗ points from the charge to the observer [2, 12, 15]. The Liénard–Wiechert field E(t) of a point charge q detected by an observer at a time t is determined by the distance r ∗ , the velocity v ∗ , and acceleration a ∗ of the charge at the emission or retarded time t ∗ = t − r ∗ /c. Defining β ∗ = v∗ /c we have  ∗  1 − (β ∗)2 q n − β∗ 2 ∗ ∗ ∗ 3 4π 0 r (1 − n ·β )

  velocity field ∗  ∗   q 1 + n × n − β ∗ × a∗ . 4π 0 c2 r ∗ (1 − n∗·β ∗)3

  acceleration field

E(t) =

(2.8)

We have indicated all retarded quantities by an asterisk. Our sign convention is that of Fig. 2.2, where n∗ · v∗ ≥ 0 is the velocity component of the charge toward the observer. Equation (2.8) is valid for any motion of a single charged particle, but it holds only if the particle is a point charge. The reason is that only then can the particle’s instantaneous motion and position be described in terms of a single retarded time. In (2.8) we have explicitly identified the two contributions to the electric field of the charge. Fields of a charge that moves with a constant velocity are called velocity fields or Coulomb fields. They are also referred to as near fields because they fall off fast with the distance to the observer as 1/r ∗ 2 . Velocity fields correspond to a ∗ = 0 and are attached to and move with the charge. In the limit v ∗ = a ∗ = 0 we obtain from (2.8) q q n∗ = n, (2.9) E(t) = Erest = 4π 0 r 2 4π 0 r ∗ 2 which is just the Coulomb field of a point charge, and by omission of the asterisk we have indicated that retardation effects are absent.

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2 Production of X-Rays: From Virtual to Real Photons

In contrast, the fields produced when a charged particle is accelerated are called acceleration fields but are best known as electro magnetic radiation which separates from the charge. Acceleration fields decrease more gradually with 1/r ∗ and owing to their dominance at large distance are also called far fields. We see from the second term in (2.8) that the far field E is perpendicular to n∗ pointing from the charge at the retarded time t ∗ toward the observer. For both terms in (2.8) the E- and B-fields are perpendicular and are linked by the following relation, B(t) =

 1  ∗ n × E(t) c

.

(2.10)

2.4 Fields of a Charge in Uniform Motion: Velocity Fields In the following we consider the strength, angular distribution, and time dependence of velocity fields surrounding moving charges. We start with the case of a moving single electron and will then consider the fields of many electrons moving as a concentrated bunch.

2.4.1 Spatial Dependence of Velocity Fields of a Single Electron The limit of a stationary charge expressed by 2.9 is illustrated on the left side of Fig. 2.3. In the rest frame of the electron, the radial electric field by definition terminates on the negatively charged electron (q = −e), and there is no magnetic field so that B = 0. We now consider the case of an electron traveling down a linear accelerator with close to the speed of light. In practice, electrons in a linac rapidly reach a relativistic velocity and their further acceleration is then quite small and can be neglected to a good approximation. We assume that an electron sits at the origin of a frame K which moves away with a velocity +v from our observer frame K . We can then transform the fields from the electron rest frame K back into the observer frame K . We start in the frame K and denote all quantities in that frame by primes. The fields in the primed electron rest frame are expressed in terms of the proper distance r . If we denote the unit position vectors in the two frames as r0 and r0 , we have E =

q r 4π 0 (r )2 0

B = 0.

(2.11)

2.4 Fields of a Charge in Uniform Motion: Velocity Fields

Fields in frame of charge B=0

69

Fields in frame of observer B

|v | ~ c E

E E

E

-

-

B

EX B v

Fig. 2.3 Example of the electric and magnetic fields of a single electron in different reference frames. Left: Electric field in the frame of the electron with charge q = −e, which is the wellknown radial Coulomb field with the electric field vector pointing toward the negative electron. There is no magnetic field. Right: The same electron is assumed to move with a highly relativistic (constant) velocity |v| ≈ c relative to the stationary observer, and the fields are observed from the stationary frame. The electric field is now nearly contained in a plane perpendicular to v since the component E⊥ ⊥ v is enhanced by a factor γ according to (2.15) and the component parallel to the field is reduced by a factor 1/γ 2 according to (2.14) over the field in the rest frame of the electron. The moving charge also gives rise to a field B which lies in the plane perpendicular to v and at any point has the direction and magnitude B = (v × E)/c2 . Both the electric, and magnetic fields move with the electron, and hence there is no “radiation”

The fields of the moving charge in the observer frame K are given by E = E

B = B = 0.

and  E⊥ = γ E⊥

B⊥ = γ

v × E . c2

(2.12)

(2.13)

We now need to express the field E in (2.11) in terms of coordinates r in the frame K since the distance in the two frames is different due to relativistic effects. We have the following relationships r0 , r0  v: r0 , r0 ⊥ v: We obtain

r = γ r r = r.

E =

q E = 4π 0 γ 2 r 2 γ2

(2.14)

E⊥ =

qγ = γ E . 4π 0 r 2

(2.15)

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2 Production of X-Rays: From Virtual to Real Photons

These are our desired results for the fields of a charge moving with a constant relativistic velocity. The above equations directly compare the fields of the moving charge E  and E ⊥ seen by the observer to the fields E that the same charge would produce if it was stationary located at the same distance r in the frame of the observer. We see from (2.14) and (2.15) that the field is reduced by a factor 1/γ 2 = 1 − β 2 parallel to the propagation direction4 and enhanced by a factor γ perpendicular to v. We can write (2.14) and (2.15) in a single general expression [2, 25, 26] 

1 − β2 E(r) = (1 − β 2 sin2 θ )3/2



q r0 , 4π 0 r 2  

(2.16)

E (r)

where θ is the angle between the direction of observation r (unit vector r0 )5 and the direction of motion v, as shown in Fig. 2.4. The direction of the E-field is seen to be along r, and the field magnitude is the largest in the plane of the charge perpendicular to v. Similar to (2.16), the corresponding B-field expression is derived by use of (2.12) and (2.13) and is given by 

1 − β2 B(r) = (1 − β 2 sin2 θ )3/2



q [v × r0 ] . 4π 0 r 2 c2 



(2.17)

B (r)

The field lines are circular about v, as shown in Fig. 2.3 for the case of a single electron. The field magnitude is largest perpendicular to v in the plane that contains the charge and vanishes for v  r0 or θ = 0. The fields given by (2.16) and (2.17) are shown on the right side of Fig. 2.3 for the case of an electron moving with nearly the speed of light. The dramatic relativistic effect leads to fields that are squished along the propagation direction and are contained in an angular field cone of half width obtained from (2.14) and (2.15) as E ⊥ /E  = 1/γ . The angular distribution of the fields is shown in Fig. 2.4 for selected cases of β. As γ increases the field pattern becomes increasingly compressed along the beam direction. Figures 2.3 and 2.4 illustrate that the fields stay attached to and move with the charge. They do not “radiate” away from or separate from the charge.

This seems to contradict (2.12), stating that E = E , but one needs to realize that this assumes that the field E is expressed in terms of coordinates r in the frame K . The relativistic correction comes from expressing the field in terms of coordinates r in the frame K . 5 Note that in (2.16) the distance r is not the distance r ∗ at the retarded time t ∗ . Rather, in the actual evaluation of the retarded field expression, r ∗ has been expressed in terms of the distance r at the observation time t. 4

2.4 Fields of a Charge in Uniform Motion: Velocity Fields

(a) angular field distribution

E

(b) approx. field cone

2

B =0.8

=0.8

71

r0 v= c

E =0

q

E E

Fig. 2.4 a Angular patterns of the E- and B-fields according to (2.16) and (2.17) for a particle of charge q, traveling with a speed v = βc, for different values of β. b Approximate angular full width of the cone perpendicular to v that contains the fields, obtained from (2.14) and (2.15) as 2/γ

2.4.2 Temporal Dependence of Velocity Fields of a Single Electron The fields given by (2.16) and (2.17) express the size and direction of the fields attached to a single charge in uniform motion. A stationary observer will experience a field pulse when the charge moves by. The duration of the field pulse may be expressed in terms of the time interval t over which the field seen by the observer is large, or equivalently, as illustrated in Fig. 2.5a, by introducing the concept of an observation cone. The observation cone is simply the relativistic field cone of FWHM 2/γ , drawn from the observer position. In the field cone approximation the field amplitude is constant (flat-top) when the point charge is located within the observation cone. More accurately, the temporal field pulse has a shape that is determined by the component E  E⊥ given by E(R, t) = where

−3/2 qγ  ∗2 1 + t 4π 0 R 2 t∗ =

γv t. R

(2.18)

(2.19)

It is plotted in Fig. 2.5b, and the pulse has a FWHM value 

1 + t ∗2

−3/2

=

1 ⇒ t ∗ = 0.8. 2

(2.20)

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2 Production of X-Rays: From Virtual to Real Photons

(b) temporal field pulse

R

vt r

q

2 field cone

E|| E

2

v

observer

E

q /R 4

E

t -~ 2 R v

4

0

q 2 R

0.8 0.6

E field (units of

observation cone

2

1

1

0

)

(a) angular field cone

t =1.6vR

0.4

t = 2R v

0.2 0

-4 -3 -2 -1 0

1

2

3

4

Time (units of R / v ) Fig. 2.5 a Geometry of an incident charge q of velocity v that passes by an observer at a perpendicular distance R. Attached to the charge we show the relativistic angular patterns of the total E-field and in blue the approximate field cone of width 2/γ , according to Fig. 2.4. The red cone shown at the observer position with the same angular width is referred to as the observation cone. The observer sees a flat-top field amplitude only when the moving charge is located within this observation cone. b Field pulse magnitude created by the charge at the site of the observer according to (2.18). The FWHM of the actual pulse shown in black is given by t = 1.6R/(γ v) and is equivalent to the shown red top flat-top approximation of width t = 2R/(γ v)

When the integral over the actual temporal shape shown in black in Fig. 2.5b is replaced by aflat-top shape shown in red, we find by comparison of the integrated field ∞ distribution −∞ (1 + t ∗ 2 )−3/2 dt ∗ = 2 an equivalent flat-top width t = 2R/γ v, as expected. The peak value is E max (t) = γ q/(4π 0 R 2 ), as shown. The corresponding B-field is derived from (2.17) and has a peak value of Bmax = E max v/c2 . The fields become huge and the pulse duration extremely short for distances R of order nanometers. To obtain large fields at larger distances, say micrometers to millimeters from the charge trajectory, it is necessary to pack a large number (∼ 1010 ) of electrons into a small volume by use of a linear accelerator. A sample can then experience these large fields when the beam is either sent directly through it [21, 27] or grazes by its surface.

2.4.3 The Fields of a Relativistic Gaussian Electron Bunch The nature of the fields associated with more than one charge may be distinguished in terms of their coherent and incoherent superposition. This is facilitated by the relativistic field cone concept shown in Fig. 2.6 for two charges, separated by a distance d along the direction of motion in the frame of the observer.

2.4 Fields of a Charge in Uniform Motion: Velocity Fields

73

Coherent and incoherent field superposition ~

2 relativistic field cone

v

q

d R

observation cones t -~ 2 Rv

q R
dγ /2 sees a single field pulse as shown by the red observation cone on the right. In the latter case, the fields produced by the two charges appear to originate from a single charge 2q. The simplifying concept of relativistic field cones associated with individual charges and the complementary concept of observation cones allows us to derive the fields associated with a Gaussian distribution of many charges as illustrated in Fig. 2.7 for three cases of observer positions.

2.4.3.1

Coherent Field Superposition

In Fig. 2.7a, the observer is located far away, at a distance R σz γ = v σt γ . For typical values σz = 50 µm and γ = 104 the shown case describes observer distances of R 0.5 m. The concept illustrated in Fig. 2.6 shows that for the case of a Gaussian bunch traveling with speed v ∼ c, the field cones produced by all charges in the bunch overlap at the observer position and therefore appear to originate from a single macrocharge Ne e. The maximum total field of the bunch E(R) is then Ne -times that of the individual single charges given by (2.18). When the total bunch falls into the observation cone, as shown in Fig. 2.7a the field seen by the observer is that of a macro-charge Ne q given by

74

2 Production of X-Rays: From Virtual to Real Photons

Fields of a Gaussian electron bunch for

(a) R

(b)

z ~0.5m

2

v

z

R

R

z ~100

~ 10

4

(c) R

z

R

m

2 R~ 50 m R~ 20 m

~

2

~

2

~

2

R ~ 500 m

R ~1 m

observer

Fig. 2.7 Illustration of the observation or coherent field cones for a Gaussian electron bunch traveling along z at nearly the speed of light with γ  104 and typical rms widths σz  50 µm along z, so that γ σz = 0.5 m, and σ R  25 µm perpendicular to z. In all cases the coherent fields at the observer arise only from charges in the observation cones. We consider only the maximum fields at an instant of time when the center of the moving bunch has its shortest distance R from the observer. a All fields superimpose coherently at the sight of the distant observer. b Only a slice of the bunch contributes to the fields at a given time and with time the field strength follows the longitudinal bunch shape exp(−z 2 /2σz2 ). c Now the fields of charges in both observation cones perpendicular to z contribute to the field, and the field is zero for R = 0 and increases with R until it saturates near the edge of the bunch

E(R) =

Ne  i=1

−e γ = Ne E e for R σz γ 4π R 2  0  E e of one electron

(2.21)

The energy flow associated with the moving velocity fields is given by the magnitude of the Poynting vector, and the intensity I of dimension [energy/time/unit area] is given by (2.22) I = 0 c E 2 (R) = 0 cNe2 E e2 . The intensity scales with the square of the number of charges and therefore their field contributions add coherently. Next we consider the case in Fig. 2.7b, where σ R  R  σz γ . This important case applies to an observer that is well outside the perpendicular bunch radius σ R  25 µm but still much closer than σz γ ∼ 1 m. In this case we need to consider the Gaussian

2.4 Fields of a Charge in Uniform Motion: Velocity Fields

75

shape (see Appendix A.2.2) of the charge distribution ρ(R, t). It is described in the longitudinal time coordinate t by an rms width σt and in the transverse coordinate R by an rms width σ R according to     −Ne e R2 t2 ρ(R, t) = exp − 2 exp − 2 , (2π )3/2 σt σ R2 2σ R 2σt

(2.23)

where σt = σz /v  σz /c. The normalization is chosen so that the integral of the charge density ρ(R, t) over time t and the transverse distance R is equal to the total charge q = −Ne e of the electron bunch. The fields for σ R  R  σz γ are obtained as [27] Ne e E(R, t) = − (2π )3/2 0 c σt R





R2 1 − exp − 2 2σ R





 t2 exp − 2 . 2σt

(2.24)

We also have B(R, t) = E(R, t)/c, which may be expressed in terms of the permeability of vacuum by use of μ0 = 1/( 0 c2 ). As the bunch moves by the observer, the field strength will follow the longitudinal Gaussian charge distribution so that the amplitude of the temporal field pulse will follow the charge that is in the observation cone per unit time, i.e. the instantaneous current. Remarkably, the fields fall off with 1/R rather than with 1/R 2 for a point charge (see (2.16) and (2.17)) so that the B-field of the bunch has the same characteristic 1/R dependence as that around a current carrying wire.6 Also, the fields do not explicitly depend on the electron beam energy Ee and are completely determined by three parameters: the total charge of the bunch Ne e, the longitudinal or temporal width σt = σz /v  σz /c, and the lateral width or beam radius σ R . The field given by (2.24) is still related to the total number of charges Ne because of the well-defined Gaussian shape of the bunch. With time the number of contributing charges follows the Gaussian distribution given by (2.24), and the maximum field amplitude is eNe . (2.26) E max = − 3/2 (2π ) 0 σz R The intensity associated with (2.24) scales with Ne2 . At a given time, the fields seen by the observer are a coherent superposition of those from the charges in the observation cone. The observation cone is thus equivalent to the coherence cone of the fields. 6 In fact, (2.24) may be derived from Ampère’s law for the B-field surrounding a current carrying wire  B · dl = 2π R B = μ0 I = μ0 d(Ne e)/dt. (2.25) L

This simplification is a consequence of relativity. For the derivation see https://stohr.sites.stanford. edu/published-unpublished-research under “Homework problems associated with the Magnetism book”.

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2 Production of X-Rays: From Virtual to Real Photons

The case in Fig. 2.7c where the observer is located within the beam diameter, i.e. at a distance R  σ R from the beam trajectory, is also given by (2.24) and B(R, t) = E(R, t)/c. If the observer is positioned on the bunch trajectory R = 0, then all fields cancel by symmetry. When the observer shifts off the beam trajectory toward the edge of the bunch, the fields will steadily increase because of the increasing difference of the number of charges in the two opposite coherence cones perpendicular to the beam trajectory. The peak field occurs at R = 1.58 σ R (see Fig. 2.8).

2.4.3.2

Incoherent Field Superposition

For a stationary observer located inside the incident beam radius, we also need to consider the fields due to individual electrons that pass by at very small distances. For example, if we choose the observer site at the center of an atom, then individual electrons may pass by within the atomic radius and it is very unlikely that more than one beam electron will be present within the atomic radius at a given time. Hence the atomic observer will only see consecutive field pulses from individual electrons. The field pulses are described by (2.18) and plotted in Fig. 2.5b. The fields are huge, and the pulse duration is extremely short for distances R of order of the Bohr radius a0 = 0.0529 nm. They may in fact lead to liberation of core electrons, called field ionization or impact ionization. It is common to characterize the field strength at atomic sites relative to the distance b of the passing electron from the nucleus, where b is called the impact parameter. These fields will only arise from individual electrons in the beam since the probability of two electrons being simultaneously within an atomic distance is negligible. If we time-average over the entire pulse, several electrons, say n, may pass by the atomic observer at different times and the intensity will be I = 0 c

n 

E i2

= 0 cn E 2 for b ≈ a0 .

(2.27)

i

  incoherent In this case the fields at the atomic sites add incoherently, as indicated. We will see that this distinction between coherent and incoherent fields leads to the virtual photon spectrum associated with the velocity or Coulomb fields of a Gaussian bunch shown in Fig. 2.11.

2.4.4 Generation of Huge Field Pulses: THz Fields and Radiation The dependence of the fields on R given by (2.24) with t = 0 is shown in Fig. 2.8 for practical beam parameters that can be achieved in a linac [21].

2.4 Fields of a Charge in Uniform Motion: Velocity Fields

77

1000

B ( Tesla )

100

11

10

10

10

E (V/m)

10

1 x 10 electrons 100 fs t= 1 m R= R = 10 m

10 9

10 1 0

20 40 60 80 100 Distance from beam center, R ( m)

Fig. 2.8 Amplitude of the magnetic induction B(R) in [Tesla] and the electric field E(R) in [V/m] as a function of the perpendicular distance R from the beam center, according to (2.24) with t = 0. We have assumed 1 × 1010 electrons per pulse and a Gaussian bunch profile in all directions, with standard deviations σ R = σx = σ y = 1 µm (black) or 10 µm (gray) perpendicular to the beam and σz = 30 µm corresponding to a temporal pulse width of σt = 100 fs. The peak value of the field occurs at R = 1.58 σ R from the beam center

For low energy (tens of keV) electron beams, the absence of relativistic effects limits the achievable peak current densities. Since the bunch length in the rest frame of the electrons is not relativistically increased for low energies (l0 =l in (2.3)), the strong Coulomb repulsion limits the number of electrons per bunch or pulse. Nevertheless, owing to the larger electron than x-ray interaction cross sections pulses of tens of femtosecond length can still be used for the study of ultrafast dynamics in many systems [28, 29].

2.4.5 Frequency Spectrum of Gaussian Electron Bunches The beam parameters used for Fig. 2.8 correspond to electron bunches that have been compressed by state-of-the-art techniques [21]. More typical linac pulses have Gaussian temporal widths of a few picoseconds [30]. Since a temporal length of 1 ps = 10−12 s corresponds to a cycle frequency, measured in units [Hz] = [1/s], of 1012 Hz = 1 terahertz (THz), such pulses have an associated spectrum in the far-infrared or THz regime. Radiation of 1 THz cycle frequency has a photon energy of ω = 4.14 meV

78

2 Production of X-Rays: From Virtual to Real Photons

1

10

10

2

10

3

10

4

10

5

10

6

Photon frequency (THz) 10

2

10

3

10

4

10

5

10

6

10

7

10

8

Photon wavenumber (cm-1 ) 0.01

0.1

1

10

100

10 4

1000

Photon energy (eV) 10

2

10

1

10

-1

10

-2

10

-3

10

-4

Wavelength ( m) Fig. 2.9 Electromagnetic spectrum in the commonly used units of frequency, wavenumber, photon energy, and wavelength

and a wavelength of 300 µm.7 For convenience we show in Fig. 2.9 an overview of the electromagnetic spectrum from the THz to the x-ray region in various commonly used units. In practice, one typically links field pulses in the time and frequency domains by a Fourier transform. For a Gaussian lineshape the Fourier transform is again a Gaussian. One needs to distinguish the widths of field pulses that are Fourier transformed from the width of the typically measured intensity pulses, which correspond to squared fields. Pulse shapes and Fourier transforms are discussed in Appendices A.2 and A.4.1. Figure 2.10a compares the shape of the temporal pulse due to the velocity fields attached to a relativistic electron bunch with that of a “half-cycle” freely propagating THz pulse. The velocity field pulse is a unipolar pulse while the freely propagating THz pulse also has a negative component required for wave propagation. The latter is however weak and extends for a long time. The Fourier transforms of the two pulses are shown in Fig. 2.10b. The frequency spectrum of the e-beam pulse peaks at zero frequency, while that of the “half-cycle” pulse looks similar but has zero field amplitude at zero frequency. Usually, freely propagating “half-cycle” THz pulses are produced by laser excitation of matter [31–34]. Coherent THz radiation is also emitted by a bending magnet of a storage ring, referred to as coherent synchrotron radiation [35–38]. Higher THz frequencies may be achieved by insertion of a thin metal foil (e.g. Be) directly into a linac beam with electron bunch lengths as short as 100 fs. The foil strips off THz fields which can then freely propagate as coherent transition radiation that may be focused. 7 Care has to be exercised in distinguishing the cycle frequency ν (units Hz) which is the inverse of the time that it takes a system to evolve through a complete cycle of 360◦ = 2π from the larger angular frequency ω = 2π ν, corresponding to the shorter time it takes a system to evolve through part of a cycle defined as 1 rad = 360◦ /2π = 57.3◦ .

2.5 Weizsäcker-Williams Method: Virtual Photon Spectrum

(a) Temporal pulse

Field B(t) ( T )

t

=100 fs

e-beam photon “half-cycle”

0

-0.5

Field B( ) ( T fs )

(b) Frequency pulse

1 0.5

79

400 300

200

1/2 t =1.6 THz

100 0

0

1 2 Time t (ps)

3

0 Frequency

1 2 (THz)

3

Fig. 2.10 a Comparison of temporal shapes of a velocity field pulse produced by a relativistic electron beam (rms width 100 fs) shown in red and the corresponding freely propagating “half-cycle” THz pulse (black) [31]. The latter also has a negative weak cycle extending to longer times, required for wave propagation. b Frequency spectra of the two pulses in (a). Only positive frequencies are allowed as indicated by shading. The frequency pulses are nearly identical except that the e-beam pulse has a finite DC component at zero frequency, while the photon pulse has no DC component

For example, using 70 fs LCLS electron bunches, a focused THz field strength of 4.4 GV/m has been achieved [39]. This is comparable to the velocity field of an electron beam at a distance of 100 µm, as shown in Fig. 2.8 [21].

2.5 Weizsäcker-Williams Method: Virtual Photon Spectrum The right side of Fig. 2.3 shows that for a single charge moving with constant relativistic velocity, both the E- and B-fields are attached to and move with the charge. The fields do not separate from the charge and hence do not “radiate” away from it [1, 12]. For an ultra-relativistic charge, the electric and magnetic fields are perpendicular to the direction of charge motion v, so that E × B ∝ v, with v  c. The two fields are also perpendicular to each other, and they obey the relationship |E| = c|B|, just like an electromagnetic wave.8 When such an electron beam traverses a sample or passes by a surface at a small distance of order of microns, the velocity fields act on the sample just like an EM wave. The close resemblance between relativistic velocity fields and EM waves was first noticed by Fermi in 1924 [40] and in the mid-1930s was developed independently by Weizsäcker [10] and Williams [11] into a powerful method, now called the Weizsäcker-Williams (WW) method of virtual photons [12, 41, 42].9 By use of this formalism one can calculate the scattering and absorption of the virtual photons, Strictly, |E| = c|B| is only true for photons, but for highly relativistic electrons it is a good approximation. 9 I would like to thank Max Zolotorev (LBNL) for discussions of the WW method. 8

80

2 Production of X-Rays: From Virtual to Real Photons

taken to be real photons, by atoms. This was convenient in the early days [40] since photon cross sections were better known. Today they are conveniently tabulated by Henke et al. [43], Hubbell et al. [44] or can be downloaded from the web [45]. The central idea is that the velocity fields accompanying the charge may be thought of as a cloud of virtual photons carried by the charge. This cloud may be liberated through interactions of the charge. For example, in materials the fields and electrons may move at different speeds so that the fields are “stripped off”. The photon cloud may also be exchanged between particles. For example, the electromagnetic repulsion between two electrons may be viewed as an exchange or transfer of virtual photons between them. Similarly, when an electron interacts with a nucleon, it is not the electron which actually “probes” the nucleon but the mediating virtual photons.10 The WW method already provided a glimpse of virtual photons which through the development of QED in the late 1940s emerged as the carriers of the electromagnetic force. Prominent examples of virtual photon exchange are [12, 42, 46, 47] • Virtual photon conversion into synchrotron radiation upon interaction with a magnetic field. • Virtual photon conversion into excited atomic electrons via photoelectric excitation, known as collisional ionization of atoms. • Virtual photon conversion into bremsstrahlung radiation by scattering on the nuclear Coulomb field. • Virtual photon conversion into transition radiation upon deceleration at a dielectric boundary. ˇ • Virtual photon conversion into Cerenkov radiation when the particle outruns the attached fields. • Virtual photon conversion into real photons during positron-electron annihilation. Each relativistic charged particle carries with it a virtual photon spectrum. Through interaction, the particle may radiate an EM spectrum that closely follows the virtual spectrum.

2.5.1 Virtual Photon Spectrum of a Gaussian Electron Bunch In practice we are interested in the virtual photon spectrum of a Gaussian bunch of electrons, which corresponds to the velocity fields discussed in Sect. 2.4.3. We have seen that the distance and time-dependent velocity fields of a Gaussian bunch E(t, R) always contain a coherent component whose intensity scales with the square 10

The larger the energy and momentum transfer to the mediating virtual photons, the shorter their wavelength. Since only very short wavelength photons can “see” the constituents of the nucleon, the quarks, larger and larger accelerators are needed to see more and more details.

2.5 Weizsäcker-Williams Method: Virtual Photon Spectrum

81

of both the number of charges and the fields of the individual electrons, I ∝ Ne2 E e2 . Inside the beam radius there is an additional incoherent intensity which scales only linearly with the number of electrons, I ∝ Ne E e2 . Classically, the EM energy is stored in equal parts in the E- and B-fields, and the energy density is given by [12],   |B|2 1 1 dE 2 |B|2 = 0 |E| + = 0 |E|2 = dV 2 μ0 μ0

(2.28)

On the right we have indicated that the energy may be expressed in terms of the E or √ B fields alone since they are related according to |E| = c|B| = |B|/ 0 μ0 . In the following we shall consider the EM energy as expressed by the E-field expression.

2.5.2 Coherent Virtual Spectrum: THz Photons The coherent part of the virtual photon intensity arises from THz fields as discussed qualitatively in Sect. 2.4.3.1. The Weizsäcker-Williams method first converts the time-dependent coherent fields E(t) into frequency dependent fields E(ω) through a Fourier transform. For a Gaussian bunch, the coherent temporal E-fields have the form (2.24) or   t2 (2.29) E(R, t) = E 0 (R) exp − 2 , 2σt where Ne e E 0 (R) = − (2π )3/2 0 c σt R





R2 1 − exp − 2 2σ R

 .

(2.30)

The frequency spectrum is given by the Fourier transform (see Appendix A.4) ∞ F(R, ω) =

E(R, t) e −∞

−iωt

  √ ω2 dt = 2π σt E 0 (R) exp − , 2(1/σt )2 



(2.31)

E(R,ω)

where E(R, t) and E(R, ω) have the same dimension. The transform is again a √ Gaussian with an rms width σω = 1/σt and an amplitude E 0 (R) 2π σt . When the square of the absolute values of the fields is integrated over time and frequency, respectively, the corresponding energies are conserved according to Parseval’s theorem (see Appendix A.4), so that +∞ +∞ √ 2 2 |E(R, t)| dt = σt |E(R, ω)|2 dω = |E 0 (R)|2 π σt . −∞

−∞

(2.32)

82

2 Production of X-Rays: From Virtual to Real Photons

Using this result, we can express the energy contained in the EM field through an integration of (2.28) that involves either the temporal fields 0 |E(R, t)|2 and the volume element dV = dA cdt or the frequency fields 0 |E(R, ω)|2 and the element dV = d A cσt2 dω, where d A is an area element in the plane perpendicular to the direction of charge motion. Hence in the frequency domain we have  +∞ A −∞

dE d A cσt2 dω = dV

 +∞ 0 |E(R, ω)|2 dA cσt2 dω A −∞

 +∞ = A −∞

dNph ω dA cσt2 dω. dV

(2.33)

In the last line we have furthermore equated the EM energy density to that contained in Nph virtual photons of energy ω in dV . By rewriting |E 0 (R)|2 defined in (2.30) in terms of the fine structure constant αf = e2 /(4π 0 c)  1/137 we obtain, |E 0 (R)|2 =

Ne2  αf 2 2π 0 c σt2 R 2



 2 R2 . 1 − exp − 2 2σ R

(2.34)

By use of this relation, comparison of the last two expressions in (2.33) yields  2 d2 Nph Ne2 αf 2 2 −(r ∗ )2 /2 e−ω σt , = 1 − e 2 dA dω/ω 2π 2 σ R (r ∗ )2

(2.35)

where we have defined r ∗ = R/σ R . When (2.35) is integrated over the plane perpendicular to the moving charge (element dA), it does not converge similar to the energy of a point charge. We may, however, calculate the energy contained in a finite circular area of radius R by integrating (2.35) in cylindrical coordinates according to R 2π 0

dNph N 2 αf −ω2 σt2 r dr = e e d Adω/ω π

R/σ  R

0



2 1 −(r )2 /2 dr , 1 − e r

 I



R σR

(2.36)



where r = r/σ R . We obtain for the number of virtual THz photons per unit bandpass in the area π R 2 , Coherent virtual photons :

  dNph R N 2 αf −ω2 σt2 I = e e dω/ω π σR

(2.37)

2.5 Weizsäcker-Williams Method: Virtual Photon Spectrum 10

10

(a) 10

83

( ) R

10

R

-1

-1

(b)

G(x)

10-3 10-5

10-3

10-7 -5

10

0.1

1

10

R/

10-9 1000 0.001

100

0.010

0.100

1

10

x

R

1018

Virtual photons / unit BW / R2

(c)

h

c=

16

10

h 2

10

Ne =1 x 10

t

= 30 GeV 4 = 6 x 10 t = 1 ps R= R bmin = a0 e

1014 12

10

2

Coherent photons

Ne

f

1010

Ne

f

G

h

10 8

= b min

Incoherent photons

10 6 10 -6

h c

c

0.001

1

1000

10 6

10 9

Photon energy (eV) Fig. 2.11 a Function I (R/σ R ) defined in (2.36). b Function G(x) defined in (2.43). c Virtual photon spectrum of a 30 GeV Gaussian electron beam (γ = 6 × 104 ) of Ne = 1010 electrons with temporal rms length σt and lateral rms width σ R . The number of virtual photons is assumed to be contained within the lateral beam radius R = σ R . The coherent spectrum is given by (2.37), and the cut off energy is 4.65 × 10−4 eV. The incoherent spectrum is given by (2.42), where we have assumed a minimum impact parameter bmin = a0 = 0.0529 nm, leading to a cut off energy of 2.23962 × 108 eV

The integral I(R/σ R ) is plotted in Fig. 2.11a, and for a circular area with radius R = σ R we have I = 4.55 × 10−2 . The Gaussian spectrum for this case is plotted in Fig. 2.11c as a red curve for the listed beam parameters. The spectrum has a cut off energy  (2.38) ωc = √ 2 σt which is also indicated in the figure. For the assumed rms pulse length of σt = 1 ps the cut off energy is 4.65 × 10−4 eV ( 0.11 THz).

84

2 Production of X-Rays: From Virtual to Real Photons

2.5.3 Incoherent Virtual Spectrum: X-Ray Photons Incoherent field superposition arises from individual electrons passing by an observer at ultrashort distances as discussed in Sect. 2.4.3.2, approximated by an “impact parameter”, usually denoted b. The virtual photon spectrum as a function of energy is again obtained from the Fourier transform of the temporal field pulse of a single electron given by (2.18). The Fourier transform operation is valid because the field of a single electron is “coherent”. Using the notation b = R, the field in the frequency domain is given by [12, 41] E(b, ω) = −

e f (2π )3/2 0 c σt b



 bω . γc

(2.39)

The prefactor is the same as in (2.30), and the unitless function f (x) = x K 1 (x) contains the modified Bessel function K 1 (x) of the second kind. f (x) is unity for small x, i.e. for ω γ c/b ∞ [12]. We also have 0 f (x) dx = π/2. We can now follow the derivation of the virtual THz spectrum and obtain for a single electron   d2 Nph (b, ω) αf 2 bω = f , (2.40) 2π 2 b2 γc dA dω ω where αf = e2 /(4π 0 c)  1/137 is the fine structure constant. The number of pseudo photons scales with 1/b2 and the cut off frequency of the distribution scales with 1/b. With increasing impact parameter, the number of photons therefore strongly decreases and the spectrum shifts to lower frequency. The total pseudo photon spectrum of a relativistic electron is given by integration of (2.40) over impact parameter b bmin 2π ∞

d2 Nph (ω) αf b db = dω π dA ω

  bmin 1 2 bω f db. b γc

(2.41)

∞

Each electron therefore contributes independently and the total incoherent spectrum scales with the number of electrons in the bunch, Ne . The total incoherent virtual photon spectrum is obtained as Incoherent virtual photons :

dNph (ω) dω ω

  Ne αf bmin ω = . G π γc

(2.42)

The function G(x) =

 1 [x K 0 (x)]2 + 2x K 0 (x)K 1 (x) − [x K 1 (x)]2 2

(2.43)

2.6 Acceleration Fields: Creation of EM Radiation

85

where K 0 (x) and K 1 (x) are modified Bessel functions of the second kind, is plotted in Fig. 2.11b. The cut off frequency ωc expressed as a photon energy is given by ωc 

γ c . bmin

(2.44)

It is seen to shift to higher energy with increasing γ and 1/bmin , as expected. The incoherent spectrum is plotted in Fig. 2.11c as a black curve, and the cut off energy is indicated. In laboratory x-ray generators, non-relativistic electron beams are used to create bremsstrahlung x-rays, and the spectrum is well known to extend to the electron energy of the incident beam [48]. If we assume that the highest virtual photon energy is also determined by the beam energy Ee = γ m e c2 , we obtain bmin = /m e c = λC /(2π ) = 3.87 × 10−4 nm, where λC = 2.43 × 10−3 nm is the Compton wavelength. For the incoherent spectrum shown in Fig. 2.11 we have assumed bmin  a0 , where a0 = 0.0529 nm is the Bohr radius. The large coherent intensity is due to the fact that the wavelength exceeds the longitudinal beam size of 300 µm (1 ps), and thus the virtual photons associated with the different electrons are in phase and add coherently. As a consequence, the number of coherent virtual photons is considerably larger than the corresponding number of incoherent photons in the same beam area. As shown by the integral I plotted in Fig. 2.11a, the coherent photon cloud is largely contained in an area approximately 10 times the beam size. In contrast, the incoherent intensity falls off as 1/R 2 with the distance R from the electron positions and in practice is negligible outside the beam area. The cut off energy of the √ coherent spectrum increases for shorter electron bunch lengths σt according to 1/( 2σt ), and it does not explicitly depend on the beam energy Ee . In contrast, the cut off of the incoherent spectrum is largely determined by Ee (or γ ) and the spectrum does not depend on the electron bunch length σt . In the following section we shall discuss how the cloud of virtual photons can be spun off the charge and turned into true photons.

2.6 Acceleration Fields: Creation of EM Radiation As discussed in Sect. 2.3, the Liénard–Wiechert fields associated with a moving charge come in two categories, velocity fields that remain attached to the charge and acceleration fields which separate from the charge as electromagnetic radiation.11 Today, one of the most important applications of the Liénard–Wiechert equation (2.8) is the description of “synchrotron radiation”, which has evolved from a “waste product” and nuisance in high energy physics to its deliberate optimization

11

The electric fields and radiation patterns associated with a charge in arbitrary motion can be visualized by the real-time radiation simulator written by Shintake [49].

86

2 Production of X-Rays: From Virtual to Real Photons

(a) Storage ring, undulator - 500MHz, 1nJ/pulse

100 ps

2 ns (b) X-FEL, warm linac - 100Hz, 1mJ/pulse

100 fs

10 ms (c) X-FEL, supercond. linac - 500KHz, 500 J/pulse

100 fs

2 s Fig. 2.12 Typical electron bunch lengths and repetition rates in storage rings and XFELs. The shown electron bunch structure is also that of the emitted x-ray pulses. The listed energy is that contained in the x-ray pulses

for research around the world. In practice, one does no longer use a synchrotron12 but an electron storage ring which provides better beam stability. Synchrotron radiation is created by electrons or positrons of constant energy (a few GeV) and velocity that circle around the storage ring. The electrons are propelled forward by a microwave radiofrequency (rf) field that consists of “rf-buckets” that rotate around the ring. When the buckets are filled with many electrons, we speak of an “electron bunch”. Typical rf frequencies are 500 Mz, corresponding to a temporal separation between buckets or bunches of 2 ns, as illustrated in Fig. 2.12. In the stationary frame of an observer, storage ring bunches have a typical length of 30 mm, corresponding to a pulse length of 100 ps, a lateral cross section of about 100 µm, and one bunch contains of order 1 nC of charge (1.6 nC=1010 electrons). The total energy per x-ray pulse is about 1 nJ/pulse (see Fig. 2.12). The bunches are kept on the desired horizontal orbit by vertical magnetic fields produced at locations around the ring by dipole electromagnets. Other magnets, such as quadrupoles, are used to focus the bunches and sextupoles to correct for aberrations. Bunches in XFELs are produced in a laser photocathode gun and are then accelerated and compressed in several linear accelerator sections. 100 fs long bunches may contain a similar number of electrons (1010 ) as in storage rings with the number proportionately reduced for shorter bunches. After being sent through a long undulator consisting of permanent magnets that wiggle the bunch to produce radiation, 12

In a synchrotron, the charged particles spiral out from a center, rather than circle at a constant radius.

2.6 Acceleration Fields: Creation of EM Radiation

87

the bunch is discarded in a beam dump. The length of the emitted photon pulses is approximately the same (within a factor of 2) as the length of the electron bunches. Differences may arise from the fact that the SASE process may only involve a fraction of the bunch, as discussed below. The pulse repetition rate depends on the type of linac used. As illustrated in Fig. 2.12, so-called warm linacs have typical rep rates of 100 Hz and superconducting linacs about 500 kHz. We shall see below that the characteristics of synchrotron and XFEL radiation are also significantly different. In the following we shall discuss the creation of EM radiation in storage rings and XFELs and its properties.

2.6.1 Distortion of Field Lines: Radiation The principle of inertia states that matter continues in its existing state of rest or uniform motion in a straight line, unless that state is changed by an external force. We therefore refer to reference frames that are in a state of constant, rectilinear motion with respect to one another as inertial frames. In inertial frames, the electric field lines emanating out from a charged particle to infinity are also straight. With increasing velocity, the electric field lines in the frame of a stationary observer become squished along the propagation direction, as shown in Fig. 2.3, but they are still straight and radial, as illustrated in Fig. 2.13a. When a particle is accelerated, it moves relative to an inertial system and a distortion of the electric field lines occurs as shown in Fig. 2.13b. Here we have assumed that a particle is accelerated from a time t0 to the time t1 with an acceleration a. We picture the event in the inertial frame of the particle associated with its uniform motion (v = const.) prior to t0 . The fact that the particle has accelerated from t0 to t1 is known only within a limited area—the event horizon—since the “signal of acceleration” travels away from the particle source at the finite speed of light c. Therefore, field lines outside the shaded radius c(t1 − t0 ) around the position of the particle at time t0 are the same as before. They still point to the original location of the particle at t0 . Within the event horizon, the field lines emerging from the particle are bent and join the old straight field lines at the event horizon. The bent field lines within the event horizon have a non-radial component along the direction of acceleration a. It is this field distortion, with an E component parallel to a and traveling away from the particle with the speed of light, that we call radiation. Electromagnetic radiation is the distortion of the electric field E that is created parallel to the acceleration direction a of a charged particle. It moves away from the particle with the speed of light and the associated fields fall off with the distance r from the particle as 1/r . The calculation of the emitted acceleration fields, i.e. the synchrotron radiation spectrum, starts from considering the fields due to the motion of a single charge, which is described by the Liénard–Wiechert equation (2.8) [2]. The effect of many

88

2 Production of X-Rays: From Virtual to Real Photons

(a) Charge in uniform motion E

E

v

+

+

v=0

v~ 0)

+E1 phase

- E1

Field amplitude

Field amplitude

(a) Classical coherent wave

+ E1 n amplitude & phase uncertainty

phase

- E1 n

Fig. 3.5 a Graphic representations of the field of a classical wave and a number state |n. a Phase dependence of the amplitude of a classical wave, e.g. as a function of time at a fixed observation point. b Phase dependence of the electric field variation for a single mode number state |n. At √ any phase value, the amplitude can take a continuous range of values with an uncertainty E = E 1 n/2 given by (3.76), where we have ignored the constant contribution from the zero-point field

3.4 The Properties of Single Mode Quantum States

143

The well-defined field of a classical EM wave and that of number states n > 1 are schematically compared in Fig. 3.5. The number state field oscillates like a classical field with a frequency ω, but the phase is not defined so that the oscillations are smeared as indicated in gray shading. Figure 3.5 reveals that the field variation of a single mode number state is in stark contrast with the stable amplitude and phase of a classical wave. The number state has a large amplitude uncertainty and a completely blurred or random phase, indicated by gray shading. Note that the uncertainty of the zero-point field, given by E ZP = E 0 = E 1 /2, is smaller than that shown in Fig. 3.5.

3.4.3 Properties of Single Mode Coherent States For a given mode pk, the most general solution of the time-dependent Schrödinger equation for the Hamiltonian (3.51) is a linear combination of number states [21] | k = c0 |0k + c1 |1k + c2 |2k + . . . .

(3.77)

By choosing appropriate coefficients cn we can construct various types of states. One specific linear combination in (3.77), the so-called coherent state, is of particular

Fig. 3.6 Roy Glauber displaying a poster in his own handwriting with the states named after him. Courtesy of Volker Steger, Science Photo Library, 2006

144

3 From Electromagnetic Waves to Photons

importance because of its close resemblance to a classical EM wave. The states are also referred to as quasi-classical states or Glauber states [22] (see Fig. 3.6). Coherent states are defined by the following linear combination of number states |mk [21, 25, 26] |αk =

1 e|αk |2 /2

∞ ∞  αm 1  (αk a†k )m |0k , √ k |mk = |α |2 /2 m! e k m! m=0 m=0

(3.78)

where on the far right we have shown how the state is created from the zero-point state by use of the creation operator a†k . In the following we shall drop the mode label k for convenience, keeping in mind that we deal with only a single k mode. Here α|α = 1 and α has the meaning of a complex field amplitude whose real and imaginary parts both form continuous ranges. The states |α satisfy the right and left eigenvalue relations a pk |α = α |α

(3.79)

α| a†pk = α| α ∗ .

(3.80)

The degeneracy parameter given by (3.69) is calculated as n coh =  α|a†pk a pk |α  = n = |α|2 .

(3.81)

The probability of finding m photons in the mode is given by a Poisson probability distribution, obtained from (3.78) as4 |α|2m  nm P(m, n) = | m | α |2 = exp −|α|2 . = e−n m! m!

(3.82)

 It is normalized according to m P(m, n) = 1. Hence we can simply write a coherent state as.  P(m, n) |m. (3.83) |α = m

Figure 3.7 illustrates distributions for three values of n. We see from Fig. 3.7 that for a coherent state the occupation probability peaks approximately at m = n, and its distribution variance is also n.

For large values of n the Poissonian probability (3.82) can also be written in the approximate Gaussian form

(m − n)2 1 . exp − P(m, n) ≈ √ 2n 2πn . 4

3.4 The Properties of Single Mode Quantum States Fig. 3.7 Poisson probability distributions P(m, n) according to (3.82), representing the probability of the number of photons m in different coherent states |α containing |α|2 = n photons

145

0.4

Probability P ( m, n )

Poisson distributions n =1 n =5 n = 10

0.3

0.2

0.1

0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of photons m per mode

A coherent state consists of a Poisson probability distribution centered around n photons, and n is also the degeneracy parameter or average number of photons per mode. The presence of adjacent number states that are coupled by the lowering and raising operators a and a† in the expression (3.61) for the electric field operator cause the mean field to be finite. To see this we write the complex number α = |α| ei θ and write the field operator E as   E = E0 a pk e−iχ + a†pk eiχ .

(3.84)

We obtain S = E = α|E|α = 2E 0 |α| cos(χ − θ ) =

 n E 1 cos(χ − θ ).

(3.85)

The variance of the field is obtained as 

 E2 ( E)2 = E 02 = 1 4

(3.86)

and the field uncertainty is E = E 0 =

E1 , 2

(3.87)

146

3 From Electromagnetic Waves to Photons

(b) Single mode coherent state npk = 1

0

E = E1 cos

E = E1 cos

+E1

2

+ npk E1

E1 /2

-

(a) Classical wave

-

0

- E1

2

- npk E1

Fig. 3.8 Graphic representations of the electric field expectation value and the phase and amplitude uncertainties for two cases. a Real field of a classical EM wave given by (3.84) with a pk = a†pk = 1. b Expectation value given by (3.85) of the field of a single mode coherent state with n pk  = 1. The mean field amplitude is E 1 |α| = E 1 n pk 1/2 = E 1 . The field uncertainty is given by that of the zero-point field, E 1 /2

which is independent of n and according to (3.75) is just the field uncertainty of the zero-point field. The close correspondence between a classical field E = 2E 0 cos χ and that of the corresponding coherent state for n pk  = 1 is shown in Fig. 3.8. As expected, a classical EM wave has a well-defined phase χ and amplitude E 1 . The signal S is well defined and has the value S = E 1 cos χ, and the field uncertainty E is zero. The mean amplitude of a coherent state is given by |α|E 1 = n1/2 E 1 which for the case n = 1 illustrated in Fig. 3.8b is just E 1 . The field uncertainty is shown shaded. As the mean photon number n of the coherent state increases, so does its amplitude E 1 n1/2 , yet the field uncertainty given by (3.86) remains constant at the value E 1 /2 of the zero-point field. Hence the coherent state becomes better defined in amplitude and phase and increasingly resembles a classical EM wave. It has the minimum uncertainty in amplitude and phase allowed by the uncertainty principle. The decreasing amplitude uncertainty with increasing number of photons n pk  has been experimentally demonstrated by Breitenbach et al. [27]. A single mode coherent state is a collective state that contains a mean number of photons n. It is composed of number substates |m whose occupation number is given by a Poisson √ probability distribution which peaks at m = n. Its field amplitude is given by n E 1 , and the uncertainty of the field is independent of n and is given by the uncertainty of the zero-point field E ZP = E 1 /2. With increasing n, a coherent state increasingly resembles a classical EM wave.

3.4.4 Properties of Single Mode Chaotic States Finally we discuss single mode chaotic states. In analogy to (3.83) they can be described by a probability distribution and single mode number states according to

3.4 The Properties of Single Mode Quantum States

Planck distributions

0.5

Probability P ( m, n )

Fig. 3.9 Planck or Bose–Einstein distributions (3.89) expressing the probability of the number of photons m in different chaotic states |β with different mean photon numbers n

147

n =1 n =5 n = 10

0.4

0.3

0.2

0.1

0

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Number of photons m per mode

[21] |βk =

∞  m=0



nm |mk . (1 + n)1+m

(3.88)

In the following we shall drop the mode label k for convenience. In this case the distribution P(m, n) is the Planck probability function given by P(m, n) =

nm . (1 + n)1+m

(3.89)

It is also referred to as a Bose–Einstein, thermal or geometric distribution. Again m P(m, n) = 1, and distributions for the same three values of n as in Fig. 3.7 are shown in Fig. 3.9. We see that for a chaotic state, m = 0 always has the largest probability with a monotonic decrease with increasing m. The degeneracy parameter given by (3.69) is calculated as n coh =  β|a†pk a pk |β  = n.

(3.90)

The signal and noise are given by averages of the corresponding expectation values for the number states. We have S = E = β|E|β = 0.

(3.91)

Chaotic light cannot carry a coherent signal, and its phase is completely uncertain. The variance of the field defined by (3.71) is obtained as

148

3 From Electromagnetic Waves to Photons







( E) = β|E |β = 2 2

2

E 02

1 n + 2



E2 = 1 2



1 n + , 2

(3.92)

where we have used the notation E 1 = 2E 0 (see (3.13)). The field uncertainty is, E =

√

 2 E0

1 E1 n + = √ 2 2



1 n + . 2

(3.93)

A single mode chaotic state is a collective state that contains a mean number of photons n. It is composed of number substates |m whose occupation number is given by a Planck probability distribution which peaks at m = 0. Its field properties are similar to those of number states with n replaced by n.

3.5 Photon Modes, Density of States, and Coherence Volume In this section we come back to the issue of the normalization volume V , which arises when classical and quantum properties of fields are linked as first discussed in Sect. 3.2.5 and encountered again in Sect. 3.3.1 in the quantization of the vector potential and field given by (3.55) and (3.61). Our discussion will lead to the fundamental concepts of the modes, the density of states, and coherence volume of the EM field.

3.5.1 Photon Modes We have previously box normalized the vector potential, by use of a large cubic box of volume L 3 = V . This is a standard quantum mechnical procedure found in the literature [19, 21, 28, 29]. This normalization is somewhat awkward since the volume is left undefined and simply assumed to be large. It must therefore drop out when the expectation values of physical observables are calculated. Let us therefore take a closer look at the box normalization concept. In deriving (3.50) we have implicitly assumed that the vector potential obeys periodic boundary conditions, i.e. that it has the same value on opposite faces of the cubic box, i.e. for the x direction A(0, y, z, t) = A(L , y, z, t) .

(3.94)

For the A field given by (3.50) we have chosen the plane wave basis functions,

3.5 Photon Modes, Density of States, and Coherence Volume

149

dk d

k

2

k d

Fig. 3.10 Illustration how the three-dimensional differential volume element dk for the scattered radiation field, shown as a dotted volume, can be replaced by dk k 2 d

1 (3.95)  p eik·r p = 1, 2 , C √ √ with box normalization C = L 3 = V . The box normalization results in a discrete set of wavevectors discussed below. In contrast, a normalization of the plane wave basis functions given by (3.95) over all space corresponds to a continuous set of wavevectors. The continuum  normalization is accomplished by a Dirac δ-function normalization where C = (2π )3 in (3.95) √ [29]. √ For the field to satisfy (3.94) with C = L 3 = V , the wave vector components must be multiples of 2π/L ⎛ ⎞ n 2π ⎝ x ⎠ ny . (3.96) k= L nz One obtains a complete discrete set of transverse orthonormal basis functions and wave vectors when the numbers n x , n y , and n z run through all whole numbers n x , n y , n z = 0, ±1, ±2, ±3, .... In order to obtain the total number of states in a finite volume V we would simply sum all values of k, i.e. over all values of n x , n y , . For a box that and n z  fills all space L → ∞ we need to replace the sum by an integral, ( k )/L 3 → ( dk)/(2π )3 , but we can simply write for the differential number of states or “modes” in the interval dk, dM =

L 2π

3 dk.

(3.97)

Note that this expression is for a single polarization. As already discussed in conjunction with (3.55) there are actually two polarization states for each k state and one therefore sometimes finds the above expression multiplied by 2. We will keep this in mind for later. We can now express the three-dimensional reciprocal space volume element dk in (3.97) by use of Fig. 3.10 in terms of the absolute value k = |k| and the solid angle d = dk around a given photon propagation direction k. The shown rectangular shape of the solid angle and the box for the volume are actually unimportant, and we could have used a cylindrical cone, for example. The important thing is that the solid angle element d = dk is centered around the wavevector direction k. By expressing dk = dk k 2 d and use of k = ω/c we can restate (3.97) as follows.

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3 From Electromagnetic Waves to Photons

The number of radiation modes in the volume V , per polarization state and per solid angle d, in the range k to k + dk is given by V 2 dMk k dk. = d 8π 3

(3.98)

In the angular frequency range ωk to ωk + dωk we obtain with k 2 dk = (ωk2 /c3 )dωk dM V ωk2 = . d dωk 8π 3 c3

(3.99)

The number of modes per polarization, per unit emission angle per unit energy per unit volume is therefore given by the general expression ωk2 dM , = d dωk dV 8π 3 c3

(3.100)

which can be used to convert the number of modes for different cases. For example, if we are interested in the number of modes emitted into a solid angle of 4π, we simply multiply the right side by this solid angle.

3.5.2 Number of Modes per Unit Energy The number of modes per unit energy (dimension [1/energy]) is particularly important. It is often called the density of states. It may be defined in general for any kind of particle and by use of (3.98) may be written as ρ(E) =

V 2 dk dMk k = d. dE 8π 3 dE

(3.101)

If we include polarization for x-rays or spin for electrons, it needs to be multiplied by 2. The dispersion factor dk/dE is however different for x-rays and electrons. For x-rays there is a linear relationship between energy and wavevector E = ω =  c k, whereas for electrons there is a quadratic relationship between the kinetic energy of the electron and its wavevector Ekin = 2 k 2 /(2m e ). For photons we can evaluate dk/dE = 1/ c and state our final result as follows. The density of states of the radiation field per unit energy and per polarization mode is given by (dropping the subscripts k for convenience),

3.5 Photon Modes, Density of States, and Coherence Volume

ρ(ω) =

V V ω2 d = d. 3 3 8π c ωλ3

151

(3.102)

It has the dimension [1/energy]. From the last expression we see that within a volume V = λ3 there is one photon of energy ω per unit solid angle d = 1 and per polarization mode.

3.5.3 Number of Modes per Unit Volume Above, we have obtained expressions for the number of modes per solid angle, (3.99), and per energy, (3.101). Combining (3.101) and (3.102) we can now obtain an expression for the number of modes per unit volume V . For a given polarization we obtain ρ(ω)ω 4π dM (3.103) = = 3. dV dV λ This means that for each photon of energy ω there are 4π modes per volume V = λ3 per polarization state or 8π modes for both polarization states. At this point it is interesting to make contact with Planck’s radiation formula which gives the spectral energy density emitted by a body of temperature T . The “spectral” aspect can be defined in different ways by normalizing the energy of the radiation to the “spectral unit” of interest, e.g. the frequency range dν, the angular frequency range dω, the wavelength range dλ or the photon energy range d(ω). In the latter formulation, the radiation energy density Erad per unit volume dV and per photon energy bandwidth d(ω) for the two fundamental polarization states in the entire solid angle d = 4π is given by 1 1 ω3 8π Erad = 2 3 ω/(k T ) = 3 ω/(k T ) . B B dV d(ω) π c e −1 λ e −1

(3.104)

We see that the fundamental prefactor in Planck’s formula is nothing but the number of modes in the volume λ3 . The modes get increasingly occupied as a function of temperature, as reflected by Planck’s thermal excitation function.

3.5.4 Coherence Volume per Mode The number of modes in the volume V can also be expressed in terms of the photon coherence volume Vcoh for a single mode k, according to

152

3 From Electromagnetic Waves to Photons

Mω =

V ω2 dω V dω V λ V = 3 . d = 3 = 8π 3 c3 λ ω λ λ Vcoh

(3.105)

We have therefore identified that each k mode of the radiation field has an associated coherence volume Vcoh = Vk . For a given propagation direction k and each of the two elementary polarization states, the coherence volume per mode k is given by V pk = λ3

λ ω = λ3 . λ (ω)

(3.106)

We see that the volume is proportional to λ3 , as expected (also see Sect. 4.2.6 below). The resolving power or inverse bandwidth can be written either in a wavelength or photon energy notation λ/ λ = ω/ (ω), where λ or (ω) depends on the shape of the photon distribution function as discussed in Sect. 4.4.2. The coherence volume may be written as the product of the lateral coherence area Acoh and the longitudinal coherence length according to Vcoh = Acoh coh , where Acoh =

λ2 λ2 ω , and coh = =λ . d λ (ω)

(3.107)

We shall come back to the importance of the coherence volume in Chap. 4 where we discuss the coherence concept in more detail and its relationship with the brightness of a source.

3.5.5 Summary In the above section we have clarified the concept of “photon modes”. This concept is usually not discussed in books about x-rays since conventional x-ray sources do not have more than one x-ray photon in the same mode. We have also derived an expression for the photon density of states or more precisely the number of modes of the photon field per unit energy. This expression will be of importance in deriving the final state density ρ(E f ) which occurs in the Kramers-Heisenberg-Dirac theory for the transition rate. Finally, we have introduced the concept of the coherence volume of an x-ray beam. We will see below that the number of photons in the coherence volume or degeneracy parameter is directly related to the brightness of a source. The derived concepts are of fundamental importance in the treatment of radiation from x-ray lasers. We can summarize as follows.

3.6 Link of Classical and Quantum Properties of Radiation

153

• The modes of the quantized electromagnetic fields are defined by the wavevector directions k and two possible photon polarizations p. The so-defined modes provide a complete description since the wavevector magnitude also defines the photon energy according to ωk = c|k|. • The photon density of states is defined as the number of modes per unit energy. • The number of modes per unit volume constitutes the prefactor in Planck’s radiation formula. • The coherence volume contains indistinguishable photons. Because photons are Bosons, the coherence volume may contain an unlimited number of photons. The number of photons in the coherence volume corresponds to the number of photons in the same mode and is called the photon degeneracy parameter.

3.6 Link of Classical and Quantum Properties of Radiation Electromagnetic waves whose frequencies extend into the microwave region cause oscillations in an antenna or cavity that can be visualized on an oscilloscope. However, there are still no detectors fast enough to measure the rapidly oscillating fields of visible light with frequencies greater than 1014 Hz, or even x-rays with frequencies up to 1019 Hz, as a function of time. For x-rays one therefore uses photodetectors which are sensitive to the light intensity or the average of the square of the electric field, whose classical cycle-average is given by (3.17) or |E(r, t)|2  = 2 |E0 |2 .

(3.108)

All other properties like the energy, momentum, flux, derived in Sect. 3.2.3 depend on this quantity. Below we shall take a look at these properties from a quantum point of view and compare them to the classical expressions. To do so, we need to calculate the expectation value of the squared field operator with appropriate wavefunctions. They consist of a generalization of the single mode number states introduced above.

3.6.1 Multi-mode Number States Single-mode states correspond to a single photon energy ω = c|k|. The quantum mechanical time–energy correlation or the classical Fourier transform relationship between time and energy then demands that they are time independent. In practice, however, we are often interested in the description of beams that contain photons

154

3 From Electromagnetic Waves to Photons

with a range of photon energies and when transformed from the frequency to time domain lead to a time dependence. States which can be represented as products of the individual substates |n pk  are referred to as multi-mode number states. They are expressed as |n pk  =

 ki

|n pi ki  = |n p1 k1 |n p2 k1 |n p1 k2 |n p2 k2 ... = |..., n pk , ... , (3.109)

pi

where p1 and p2 are the two polarization basis states. The action of the creation and annihilation operators on such multi-mode states is restricted to the specific mode pk of the operator. It raises or lowers only the occupation number of one specific mode by unity, while leaving all other occupation numbers unchanged, according to √ n pk |..., n pk − 1, ...  † † a pk |n pk  = a pk |..., n pk , ... = n pk + 1 |..., n pk + 1, ... .

a pk |n pk  = a pk |..., n pk , ... =

(3.110)

We have n pk |a pk a†pk |n pk  = n pk + 1 n pk |a†pk a pk |n pk  = n pk ,

(3.111)

and therefore the sum rule  pk

! N pk

!





†

a pk a pk n pk = n pk = n ph , = n pk

pk

pk

(3.112)

where n ph is the total number of photons. We shall now use these states to calculate the quantum mechanical expectation values for the key classical properties of photon beams such as the energy, momentum, intensity, and flux discussed in Sect. 3.2.3.

3.6.2 Expectation Value of the Squared Electric Field Owing to the inability of measuring the EM field directly at optical and higher frequencies, the key quantity that is related to the measured intensity is the expectation value of the absolute value squared of the electric field |E(r, t)|2 . In calculating the quantum mechanical expectation value we use the expression of the electric field operator given by 3.61. The quantum mechanical expectation value of the square of the electric field is calculated with the multi-mode number states according to |E(r, t)|2  = n pk | |E(r, t)|2 | |n pk  .

(3.113)

3.6 Link of Classical and Quantum Properties of Radiation

155

We obtain

"

√  ωk1 ωk2 n pk

n pk | |E(r, t)| |n pk  = 20 V uv k1 k2    u auk1 ei(k1 ·r−ωk1 t) +  ∗u a†uk1 e−i(k1 ·r−ωk1 t) # 

 i(k2 ·r−ωk2 t) ∗ † −i(k2 ·r−ωk2 t) +  v avk2 e ·  v avk2 e

n pk . 2

We have explicitly indicated the dot product between the two square brackets. This expression can be evaluated with the rules given by (3.110). Each of the terms in the sum acts on only one of the substates of |n pk  at a time, and only the diagonal matrix elements contribute to the sums. The two non-vanishing matrix elements each contain a product of a creation and annihilation operator and are given by (3.111). The phases associated with the non-vanishing terms fall out and the dot products of the unit polarization vectors are unity. We then obtain

  1  1 (2n pi ki +1) ωki = ωki . n pi ki + 20 V i 0 V i 2 (3.114)  By subtracting the energy of the vacuum state 21 i ωki we obtain n pk ||E(r, t)|2 |n pk  =

n pk ||E(r, t)|2 |n pk  =

n ph ω 1  n pi ki ωki = . 0 V i 0 V

(3.115)

To obtain the final expression in terms of the total number of photons n ph , we have used (3.112) and assumed that all n pk photons that may be in different modes i have the same energy ω. We are now in a position to relate the classical amplitude of the electric field given by (3.17) and the quantum mechanical amplitude given by (3.115). The expectation value of the electric field of an EM wave can be expressed by linking the classical and quantum mechanical expressions according to |E(r, t)|2  = 2|E0 |2 =

n ph ω , 0 V

(3.116)

where n ph is the total number of photons of energy ω contained in a volume V. We can readily write down the following relations between the classical and quantum mechanical expressions. The energy contained in the field, classically given by (3.22), is

156

3 From Electromagnetic Waves to Photons

Energy: E = 20 V |E0 |2 = n ph ω.

(3.117)

We note that we could have obtained the total energy directly as the expectation value of the Hamiltonian (3.54).5 The classical power (3.23) becomes Power: P0 = 20 c A |E0 |2 =

n ph ω . t

(3.118)

For the momentum of an EM wave, classically given by (3.24), we obtain Momentum: P =

n ph ω 2 0 V |E0 |2 = = n ph  |k|. c c

(3.119)

Similar to the photon energy, we could have obtained the photon momentum as the expectation value of the photon momentum operator.6 The intensity, classically given by (3.25) is given by Intensity: I0 = 20 c |E0 |2 =

n ph ω c . V

(3.120)

The photon flux, classically given by (3.26) is Flux: 0 =

n ph c 2 0 c |E0 |2 = . ω V

(3.121)

Finally, we obtain for the fluence, whose classical expression is (3.27) Fluence: F =

5

n ph ω 2 0 V |E0 |2 = . A A

(3.122)

The total energy of the radiation field is E = Hrad  = n pk |Hrad |n pk  = n pk | =





ωk a†pk a pk |n pk 

pk

ωk n pk = n ph ω.

pk 6 The photon momentum operator, normalized to the vacuum momentum, is given by P =  † pk k a pk a pk so that P = n ph |k|.

References

157

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35.

P.A.M. Dirac, Proc. Roy. Soc. A 114, 243 (1927) P.A.M. Dirac, Proc. Roy. Soc. Lond. A 114, 710 (1927) B.J. Hunt, Phys. Today 65, 48 (2012) J. Stöhr, H.C. Siegmann, Magnetism: From Fundamentals to Nanoscale Dynamics (Springer, Heidelberg, 2006) A. Kirilyuk, A.V. Kimel, T. Rasing, Rev. Mod. Phys. 82, 2731 (2010) J.D. Jackson, Classical Electrodynamics (Wiley, New York, 1999) D. Attwood, A. Sakdinawat, X-Rays and Extreme Ultraviolet Radiation: Principles and Applications, 2nd edn. (Cambridge University Press, Cambridge, 2017) J. Als-Nielsen, D. McMorrow, Elements of Modern X-Ray Physics, 2nd edn. (Wiley, New York, 2011) E. Hecht, Optics, 4th ed. (Addison-Wesley, Reading, 2001) A. Hofmann, The Physics of Synchrotron Radiation (Cambridge University Press, Cambridge, 2004) M. Faraday, Phil. Trans. Roy. Soc. Lond. 1, 136 (1846) L. Pasteur, Annales Chimie Physique 28, 56 (1850) J. Stöhr, NEXAFS Spectroscopy (Springer, Heidelberg, 1992) G.N. Lewis, Nature 118, 874 (1926) A. Einstein, Ann. Phys. 17, 132 (1905) E. Fermi, Rev. Mod. Phys. 4, 87 (1932) M. Planck, Verh. Deut. Phys. Ges. 13, 138 (1911) P.W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics (Academic Press, San Diego, 1994) L.I. Schiff, Quantum Mechanics, 3rd edn. (McGraw-Hill, New York, 1968) F. Mandl, G. Shaw, Quantum Field Theory (Wiley, New York, 1993) R. Loudon, The Quantum Theory of Light, 3rd edn. (Clarendon Press, Oxford, 2000) G. Grynberg, A. Aspect, C. Fabre, Introduction to Quantum Optics, From the Semi-Classical Approach to Quantized Light (Cambridge University Press, Cambridge, 2010) D.A. Steck, Quantum and Atom Optics. Available online at http://steck.us/teaching (2017) R. Glauber, Optical coherence and photon statistics. in Quantum Optics and Electronics, ed. by A.B.C. de Witt, C. Cohen-Tannoudji (Gordon and Breach, New York, 1965) R.J. Glauber, Phys. Rev. 130, 2529 (1963) M. Fox, Quantum Optics: An Introduction (Oxford University Press, Oxford, 2006) G. Breitenbach, S. Schiller, J. Mlynek, Nature 387, 471 (1997) A. Messiah, Quantum Mechanics (Wiley, New York, 1958) J.J. Sakurai, Modern Quantum Mechanics, Revised (Addison-Wesley, Reading, Mass., 1994) A. Fresnel, Œuvres 1, 731 (1822) A. Einstein, O. Stern, Ann. Phys. 40, 551 (1913) J. Mehra, H. Rechenberg, Found. Phys. 29, 91 (1999) C.C. Gerry, P.L. Knight, Introductory Quantum Optics (Cambridge University Press, Cambridge, 2005) M.O. Scully, M.S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997) P.A.M. Dirac, Quantum Mechanics, 4th edn. (Oxford University Press, Oxford, 1958)

Chapter 4

Brightness and Coherence

4.1 Introduction In modern x-ray science, the concept of brightness or brilliance plays an important role as a measure of the quality of x-ray sources. We intuitively associate brightness with the intensity that falls into the eye of an observer. Today, the term “bright” or “brilliant” is even used in different connotations, like the description of exceptional intellectual abilities.1 One goal of this chapter is to provide a rigorous discussion of the brightness of a light source, in general, and of the more specific concepts of average brightness used to describe synchrotron radiation (SR) and peak brightness utilized for the description of XFEL sources. It is well known that the improvement in brightness of SR sources is associated with increasing coherence. The detailed link of brightness and coherence, however, is not obvious since we typically define “coherence” as the ability of waves to create an interference pattern, while we associate “brightness” with photon density. Hence the connection between brightness and coherence goes to the very heart of the waveparticle duality of light and touches on Dirac’s famous and puzzling statement that each photon can only interfere with itself. The elucidation and connection between the seemingly different concepts of brightness and coherence is a key goal of this chapter. In particular, we will show that the conventional concepts of brightness and coherence are equivalent descriptions of light in the first order of the fundamental theory of light and matter, quantum electrodynamics (QED). Before the advent of XFELs, this first order description of light was all that was needed in x-ray science. The reason is that even the most advanced synchrotron radiation sources produced less than a single photon within a source volume of order λ3 as illustrated in Fig. 1.3. Hence 1 When the concept of brightness was first introduced to support the proposal for third generation synchrotron sources in the USA, there was a funny exchange between a scientist in the audience and the speaker (both shall remain unnamed). The question “who needs brightness?” was answered “Well, it’s a good thing to have. Some of us have it and some of us don’t”.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_4

159

160

4 Brightness and Coherence

the question of correlations between photons, e.g. whether they can work together or interfere, did not have to be addressed. This is what is behind Dirac’s famous statement that applied to the light sources available at his time. Figure 1.3 also shows that a new way of thinking is required when considering science with XFELs since an incredible number of up to 1010 photons are born within the minimum coherence volume of order λ3 . This requires an improved description of the nature of light, called quantum optics, which was developed in the 1960s in the wake of the formulation of QED and the invention of the maser and laser. Quantum optics extends Dirac’s first order formulation of the quantum properties of light, based on non-interacting individual photons, through an infinite series of photon-photon correlations underlying QED. In this chapter we shall only discuss the conventional or first order description of light, in which the wave and quantum concepts yield the same results. This allows a “semi-classical” description of light where, based on the wave-particle duality, different experiments may be conveniently explained in either the wave or photon picture. The key to the first order quantum treatment is that interactions between photons are neglected. They emerge in a second order and higher order treatment of coherence that will be discussed in Chap. 5.

4.2 Brightness and Coherence 4.2.1 Introduction to the Concept of Brightness The concept of spectral brightness, B, was first used in radiometry and astronomy to describe the brightness of stars. Sources that emit isotropically are referred to as Lambertian sources after Johann Heinrich Lambert who in 1760 considered the angular emission properties of primary and secondary (i.e. scattering) sources. It can be shown that thermal or blackbody sources are Lambertian [1]. The total energy E = n ph ω per spectral interval (ω) emitted by such sources is given by E = B d A dt d, (ω)

(4.1)

where the proportionality factor B is called spectral brightness or spectral radiance. It is a fundamental measure of how bright a star appears to an observer at a point on earth. Here d A is the unit source area, dt the unit time of observation, and d the solid angle element of emission or observation. Since stars are typically envisioned as spheres, they emit uniformly into the 4π solid angle. If the radiating surface is flat, the emission of a Lambertian source is no longer isotropic but has a cos θ distribution around the surface normal, known as Lambert’s cosine emission law. The intensity is maximum in the forward direction and has a node for θ = ±90◦ . For a flat Lambertian source, (4.1) changes to [2]

4.2 Brightness and Coherence

161

E = B d A dt d cos θ, (ω)

(4.2)

where θ is the polar angle of the emission direction from the surface normal of the flat radiating surface. The so-called obliquity factor cos θ accounts for the projection of the flat surface seen by an observer at angle θ from the surface normal. Because of the narrow emission cone of synchrotron radiation sources around the forward direction, where cos θ = 1, the brightness defined in (4.1) is a true measure of the intrinsic quality of such sources. The brightness concept was introduced for the description of synchrotron radiation by K.-J. Kim in 1986 [3] to describe the improvement provided by third generation sources.2 For synchrotron radiation sources we can write the brightness in terms of different quantities as B=

n ph ω n ph ω n ph ω 1 1 1 = = dt (ω) As d dt (ω) As d dt As d (ω)         

spectral power Pω

spectral intensity Iω

intensity I

n ph n ph 1 . = = dt A  (ω)/ω d As dt (ω)/ω   s     flux φ

(4.3)

photons in phase space volume

The brightness expression is based on the concept of space-time separability since it factors into lateral or spatial properties expressed through As d and longitudinal or temporal properties expressed through dt (ω)/ω. This separability allows the distinction of lateral or spatial coherence and longitudinal or temporal coherence. The last expression furthermore links brightness to the number of photons per phase space volume. The understanding of the link of brightness, coherence, and photon density is a key goal of the present chapter, with emphasis on an x-ray perspective of synchrotron and XFEL radiation.

4.2.2 Formal Definition of Brightness In x-ray science, the most-used concept is the photon flux or number of photons per unit area per unit time and the related intensity which is the photon flux times the photon energy, as discussed in Sect. 3.2.3. Spectral brightness or brilliance is a more fundamental property that describes the quality of an x-ray source and the properties of the emitted radiation. We can picture the x-rays emitted from an electron bunch as illustrated in Fig. 4.1. 2

This concept greatly helped in making the case for the construction of the Advanced Light Source (ALS) and the Advanced Photon Source (APS) in the USA. Relative to the four orders of magnitude increase of x-ray brightness, the modest flux increase offered by the competing proposal for the construction of a spallation neutron source led to the higher priority funding selection of the x-ray sources.

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4 Brightness and Coherence

Fig. 4.1 Illustration of the spectral brightness concept to the lateral (red) and longitudinal (blue) properties of a source. The peak brightness is defined as the number of photons n ph per source size As , per solid angle of emission d (units [steradian]), per temporal pulse length τs , per relative bandwidth ω/ω. In practice, the area–solid angle product As d, and the orthogonal time-bandwidth product τs ω/ω are determined by both electron and photon distribution functions. Underneath we have identified the diffraction and transform limits for a coherent source. The factors H and K depend on the shape of the intensity distribution of the source and are given by (4.19) and (4.20), below

For simplicity we have drawn the source volume resembling an ellipsoid of lateral area As and length s . In general, the source volume may be expressed as a product of an effective area and length that contain the generated energy, which leads to the shape dependent factors H and K in the stated diffraction and transform limits in the figure. The lateral source area As and the solid angle of emission d define the transverse source properties.3 The solid angle d, illustrated in Fig. 4.2, has units of steradian (sr), so that the maximum solid angle of emission from a point is 4π sr. A light source has a minimum coherence area of  λ2 , and emission into the cone d  1 sr corresponds to coherent emission. The source length s defines the x-ray pulse length which is taken to be the length of the electron bunch. The uncertainty in the length of the emitted wavevector or wavelength λ = 2π/k determines the relative bandwidth. The uncertainty in photon wavelength λ and photon angular frequency ω = 2π ν and energy (ω) is linked according to λ = (c/ν) = cν/ν 2 . The relative bandwidth may be written equivalently as k λ ω (ω) τ0 . (4.4) = = = = k λ ω ω τcoh 3 The solid angle of unit [sr] and the planar angle of unit [rad] are defined as ratios of two areas or lengths and are actually dimensionless. The names are introduced to distinguish between dimensionless quantities of different nature.

4.2 Brightness and Coherence

163

r

solid angle

total planar angle =65.5o

d =1sr area

r2

Fig. 4.2 Illustration of a cone of solid angle of d = 1 in units of steradian (sr), corresponding to the coherence cone of a point source. In practice, d = 1 can be considered the coherence cone of a source of minimum coherence area λ2 . Also shown is the associated total planar opening angle ϑ  65.5◦ . In general, the solid angle d is related to the total planar opening angle ϑ about the central axis according to d = 4π sin2 (ϑ/4), which for small angles becomes d  π(ϑ/2)2

where τ0 = λ/c, and the wavelength and wavevector are related according to λ = 2π/k. The spectral brightness, identified by the boxed expression in Fig. 4.1, contains the fundamental uncertainty relations in space, identified in red, and time, shown in blue. Owing to the transverse nature of light it assumes separability of the lateral or spatial and longitudinal or temporal degrees of freedom. With these considerations we can define as follows. The spectral brightness of a photon source is defined as B=

n ph c = As d s λ/λ

n ph 

As d  

space-momentum uncertainty

τ ω/ω  s  

.

(4.5)

time-energy uncertainty

Here n ph is the number of photons. The spatial part is determined by the lateral source area As , and the dimensionless solid angle of emission d (in sr). d = 1 (in sr) defines the coherence cone of a source of area λ2 , which has a total planar opening angle of ϑ  66◦ as shown in Fig. 4.2. The time τs is the temporal duration corresponding to a length s = c τs , and ω/ω = λ/λ is the dimensionless relative bandwidth of the emitted radiation. By convention, the spectral x-ray brightness is stated by use of the small angle approximation d = π ϑ 2 /4 in units of [photons/s/mm2 /mrad2 /0.1% bandwidth], where the correspondence between d with units [sr] and the total planar opening angle ϑ with units [rad] is illustrated in Fig. 4.2.

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4 Brightness and Coherence

By use of (4.4) and c/ s = 1/dt, the spectral brightness definition (4.5) agrees with the radiological one given by (4.3). We see that spectral brightness is defined abstractly as the number of photons in a six-dimensional phase space volume, comprised of a three-dimensional spatial volume V = As s that contains the photons, the two-dimensional solid angle d into which photons are emitted and the onedimensional bandwidth λ/λ of the radiation. Spectral brightness is conserved in phase space, which is a general consequence of Liouville’s phase space theorem. For example, through focusing one can create a smaller image of the source size As , but this comes at the expense of the divergence angle d, since their product is constant. Another example is the reduction of the emitted bandwidth ω by use of a monochromator at the expense of an increased pulselength τs . Spectral brightness of a source is a conserved quantity in phase space and cannot be improved by optical elements. Imperfect optics will, however, lead to its degradation. We note upfront that the so-defined concept of brightness is only a first order concept in the ultimate theory of light, QED, as will become apparent later.

4.2.3 Average Brightness In (4.5) we have identified by underbrackets two orthogonal phase space contributions. Assuming x-ray propagation in the z direction, the first bracket defines the “lateral” source properties given by the lateral source area As , typically specified by the lengths along two perpendicular axes x and y, defined to lie in the ring plane and perpendicular to it. The solid angle of emission is then expressed as d = π θx θ y , where θx and θ y are the half planar emission angles measured from the x and y axes in the two orthogonal x-z and y-z planes, respectively. The product As d defines the emittance of the source which is the key property of storage ring sources. From a SR perspective, the time-bandwidth part of the brightness expression indicated by the second underbracket in (4.5) is simply averaged and the so-called average brightness is defined as Bavg =

n ph F , where F = . As d t ω/ω

(4.6)

Here F is the spectral photon flux with dimension [photons/time/bandwidth], to be distinguished from the areal photon flux with the dimension of [photons/time/area]. The average brightness Bavg of a storage ring can be increased by a limited amount through reducing the time–energy uncertainty in (4.5). This is done by making each

4.2 Brightness and Coherence

165

electron radiate longer wavetrains by sending them through an undulator. In practice, undulators reduce ω by a factor of about 100 relative to a bending magnet, with an increase of F by the same amount. Since most storage rings have similar cycle repetition rates R  500 MHz and pulse lengths τp  100 ps, one defines the emission time t in units of [s] as the time over which photons are emitted by the electron bunches per ring cycle, t  5 × 10−2 s, and by convention one assumes ω/ω = 10−3 (0.1%). As indicated by underbrackets in (4.5), the achievable lateral or average brightness is ultimately determined by uncertainty relations. In storage ring sources, the minimum time-bandwidth product τs ω/ω can only be reduced (without loss of flux) by narrowing the bandwidth ω through undulators. The real improvement arises through the reduction of the product As d, i.e. through smaller beam emittances [4]. Owing to the fact that the area-angle product enters in the denominator, the brightness has an upper bound. Its limit is given by Heisenberg’s space-momentum uncertainty principle, and when this limit is reached, the source becomes diffraction limited and all emitted photons are laterally coherent. Shortly after Heisenberg’s publication of the space-momentum uncertainty principle [5], Kennard [6] first proved the modern inequality which links the standard deviation of position, r , with that in momentum, p. For this reason, Gaussian intensity distributions of rms width σ I naturally define the minimum uncertainty distributions in position, r and momentum, p, which can also be written in terms of the wavevector, q = p/, and angle θ = q/k = qλ/2π according to σrI σ pI ≥

 1 λ , or σrI σqI ≥ , or σrI σθI ≥ . 2 2 4π

(4.7)

For coherent Gaussian intensity distributions with rms widths σrI and σθI , the first underbracket in (4.5) is given by Gaussian diffraction limit : As d = 2π(σrI )2 2π(σθI )2 ≥

 2 λ . 2

(4.8)

The minimum of the product, expressed by the equal sign, defines the so-called diffraction limit according to Acoh dcoh = λ2 /4 (see Fig. 4.1). This expression means that a source area of λ2 /4 emits into a diffraction limited solid angle coh = 1. When the diffraction limit is reached, all n ph photons are laterally coherent and the average brightness has its maximum max = Bavg

4F . λ2

(4.9)

In practice, only a fraction of the photons emitted by today’s storage rings are laterally coherent due to the fact that the lateral size of electron bunches in storage rings cannot be compressed below the uncertainty area associated with a single

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4 Brightness and Coherence

electron illustrated in Fig. 2.23b. Such a source may be described to have a laterally quasi-homogeneous intensity distribution [7], where the intensity changes rather homogeneously with distance while the coherence changes much faster. If we denote μ the single-electron coherence area as Acoh and the total lateral beam area as As , the average brightness (4.6) is reduced by the laterally coherent fraction μ

hc =

Acoh As

(4.10)

and becomes μ

Bavg =

Acoh max B . As avg

(4.11)

The photon energy dependence of the laterally coherent fraction is plotted in Fig. 2.24d.

4.2.4 Peak Brightness The second underbracket in (4.5) defines the “longitudinal” or temporal source properties, i.e. those along the beam direction. It has become of prime interest with the advent of XFELs where all photons are crammed into an ultrashort pulse of τs  1– 100 fs duration. One now uses the actual pulse duration and refers to B as the peak brightness. The reduction of the time-bandwidth product τs ω/ω at XFELs over storage rings leads to a much larger value of the peak relative to the average brightness. In analogy to the minimum area-angle relation (4.7), the corresponding minimum time-frequency Gaussian intensity distributions are given by σtI σEI ≥

σIc  1 λ , or σtI σωI ≥ , or σtI ω ≥ . 2 2 ω 4π

(4.12)

The second underbracket in (4.5) can then be written as ω √ τs = 2π σtI ω

√

σI 2πσωI λ = 2π σtI E ≥ . (4.13) ω ω 2c √ By relating the Gaussian pulse length τs = 2π σtI and effective energy √ effective width E = 2πσωI we have Gaussian transform limit : τs E ≥ h/2 .

(4.14)

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167

The minimum product, given by the equal sign, is now expressed as a time-bandwidth product that defines the so-called transform limit (see Fig. 4.1). When it is reached, all n ph photons have become temporally coherent. The expression (4.13) states that a minimum pulse length of τs = λ/(2c) emits a spectrum of unit bandwidth ω/ω. With increasing coherent pulse length the emitted bandwidth ω/ω is proportionally reduced. The peak brightness for a source with spatiotemporal Gaussian distributions is given by Bpeak =

8 n coh n ph ph c max = Bpeak . ≤ As d τs ω/ω λ3

(4.15)

The maximum value of the peak brightness is expressed by the equal sign. It corresponds to the minimum spatiotemporal uncertainty product where all photons are (first order) coherent n ph = n coh ph . The number of coherent photons in this limit is the photon degeneracy parameter to be discussed in more detail in Sect. 4.2.7 below.

4.2.5 Brightness Reduction Through Partial Coherence In practice, conventional SASE and even self-seeded pulses are not temporally coherent or transform limited and are said to be partially coherent. In general, all sources that are not fully coherent fall under the category of “partial coherence”. This difficult and broad concept covers all kinds of light behavior between the extremes, referred to as completely coherent and incoherent or chaotic. We will come back to the description of chaoticity below and at this point illustrate partial coherence through the properties of SASE XFEL pulses. A macro-pulse of a SASE XFEL of total length τp = τs consists of M coherent μ microspikes or modes of duration τcoh as previously illustrated in Fig. 2.31, where μ M ∼ τp /τcoh . In this case, the peak brightness is reduced by the temporally coherent faction Temporal coh. fraction : kc =

μ τcoh 1  . τp M

(4.16)

For example, a SASE pulse of τp  100 fs consists of M  100 spikes, so that the peak brightness is reduced from its transform-limited value by a factor of  100. The behavior of partially coherent sources in time and space is illustrated in Fig. 4.3. In Fig. 4.3a we illustrate the concept of partial temporal coherence. We have assumed a total pulse length τp , defined by the statistical average over many pulses, indicated by a smooth envelope function. Each pulse consists of a spiky substructure μ of average width τcoh indicated in red. The pulse contains a temporally coherent μ μ fraction of photons τcoh /τp . In the limit τcoh  τp the source is called temporally

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4 Brightness and Coherence

(a) Partially coherent pulse I (t)

I( ) coh

coh p

0

t coh

p=

p

p

0 coh

>h 2 -

(b) Partially coherent source area reciprocal space image

real space source

d

Acoh

s

d

coh

coh

> -

As

Acoh d

s=

As d

2

4

Fig. 4.3 a Illustration of the intensity distribution in a pulse emitted by a temporally partially coherent source. The macroscopic temporal intensity distribution in the pulse of length τp consists of μ microscopic coherent spikes of length τcoh < τp , and the pulse has a similar structure in the energy domain. The complementary uncertainty product relationship in the time and energy domain is μ μ τcoh Ep = τp Ecoh ≥ h/2 underneath, assuming Gaussian lineshapes . b The equivalent concept of a spatially partially coherent source. The macroscopic intensity distribution across the source μ area As contains microscopic coherent regions of size Acoh < As . The complementary uncertainty product relationship in the spatial and angular domain is shown underneath, assuming effective Gaussian areas μ incoherent or chaotic, and in the limit τcoh = τp it is coherent. Underneath we have indicated the time–energy uncertainty relation (4.14) for Gaussian lineshapes. In Fig. 4.3b the corresponding concept of partial spatial coherence is illustrated. When averaged over a pulse length, the source has an area As that contains microμ coherence regions of size Acoh , indicated by red circles. For simplicity we have assumed all regions to have circular shapes. The source then contains a spatially coherent fraction of photons, given in analogy to the temporal coherent fraction (4.16) by, μ

Lateral coh. fraction : h c = μ

Acoh As

(4.17)

In the limit Acoh  As the source is called spatially incoherent or chaotic, and in μ the limit Acoh = As it is coherent. At the bottom of Fig. 4.3b, we have indicated the area–solid angle uncertainty product (4.8) by assuming effective Gaussian areas.

4.2 Brightness and Coherence

169

Measured SASE pulse correlation in energy and time:

coh

p~

p

coh

coh ~1 p ~10

p

~10

coh ~1

Fig. 4.4 Measured SASE pulse structures in energy (top row) and time (bottom row) for three exemplary shots [8]. The XFEL pulses of ∼ 7 fs FWHM and a nominal photon energy of 1180 eV had similar total energies of from left to right 36.5 µJ, 39.5 µJ, and 40 µJ. In the left column we have superimposed approximate red and blue curves on the original data to indicate the boxed relationship given in Fig. 4.3a

The peak brightness of a partially coherent source is therefore reduced according to max . Bpeak = kc h c Bpeak

(4.18)

The correlation of temporal and energy widths illustrated in Fig. 4.3a are born out by experimental results by Hartmann et al. [8] shown in Fig. 4.4. In the left column of the figure we have indicated the relationship between the temporal and energy widths of the SASE structure by superimposing approximate red and blue curves on the original data shown as black curves. The temporal structure of the SASE pulses was obtained with attosecond resolution via angular streaking of Ne 1s photoelectrons produced by the x-rays. The photoelectrons are modulated by the continuously rotating electric field of a circularly polarized infrared (λ = 10.6 µm) laser, and depending on the amplitude and phase of the streaking field at the time of ionization, the photoelectrons are either accelerated or decelerated. The induced momentum change results in modulations of the photoelectron energy and angular distribution that is measured with an array of time-of-flight detectors. It is important to realize that the corresponding patterns illustrated in Fig. 4.3 in conjugate variables of time–energy and space–reciprocal space are not linked by a simple Fourier transform. The reason is that they correspond to intensity patterns where the phase information that is contained in the fields has been lost.4 4 The case shown in Fig. 4.3b is related to the well-known “phase problem” in x-ray crystallography where the reciprocal space intensity diffraction pattern cannot simply be Fourier transformed into a real space image as discussed in Sect. 8.8.

170

4 Brightness and Coherence μ

μ

With the introduction of the coherent fractions τcoh /τp and Acoh /As in the temporal and spatial domains, we can now link brightness and coherence more generally.

4.2.6 Link of Brightness and Coherence The peak brightness expression (4.15) assumes Gaussian distributions in space and time. In particular, the prefactor 8 is due to the expressions for the minimum Gaussian space-momentum uncertainty product or diffraction limit expressed by (4.8) and the minimum Gaussian time–energy uncertainty product or transform limit (4.14). Gaussian distributions are convenient because the Fourier transform of a Gaussian is also a Gaussian and from a quantum mechanics point of view they define the minimum uncertainty product. They are also typically used to characterize the temporal and spatial distributions of synchrotron and XFEL radiation. More generally, one may define shape-dependent dimensionless coherence factors, where H is determined by the functional form of the lateral (spatial) emission profile and K by the longitudinal (temporal) profile. The expressions leading to the diffraction and transform limits are then given by the following generalized forms, previously stated in Fig. 4.1. The diffraction limit for a source of arbitrary intensity distribution function is given by the minimum value of the area–solid angle product As d ≥ H λ2 .

(4.19)

The transform limit is the minimum value of the so-called time-bandwidth product

s

λ ≥ K λ or τs E ≥ K h, λ

(4.20)

= E (see (4.4)) and h = 4.136 eV fs is where we have used s = τs c and λ λ ω Planck’s constant. The coherence factors H and K depend on the shape of the longitudinal and transverse intensity distribution functions in the source, given below by (4.25) and (4.24).

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171

We can then write down the following generalized expression for the peak brightness. If we define the coherence volume through shape-dependent dimensionless lateral (spatial) and longitudinal (temporal) coherence factors H and K as Vcoh = H K λ3 ,

(4.21)

then the photons born in Vcoh are emitted into a unit solid angle d = 1 and have unit bandwidth ω/ω = 1. Defining the laterally and temporally coherent fractions of photons by factors μ

hc =

μ

Acoh τ , and kc = coh , As τp

(4.22)

where the pulse length is also the source length, τp = τs , the master equation for peak brightness reads Bpeak =

h c kc n ph c . H K λ3

(4.23)

The right side shows that the peak brightness of a source reflects the number of coherent photons n coh = h c kc n ph of wavelength λ that are born in the coherence volume Vcoh = H K λ3 and emitted into a unit solid angle with unit bandwidth. Brightness therefore reflects the number density of coherent photons of a given wavelength λ or alternatively the energy density of the distribution of photons of energy ω. The parameters H and K depend on the shape of the spatial and temporal distribution functions and definition of their widths. In this book we will follow the definition of distribution widths used in statistical optics [9]. They are based on a powerful theorem involving the absolute values squared of a Fourier transform pair known as Parseval’s theorem which assures conservation of energy associated with a Fourier transformation as discussed in Appendices A.4.1 and A.4.4 (also see Sect. 4.4.2 below). For coherent sources, the factors H and K follow from Fourier relations between the field distributions. For incoherent sources, the factors are calculated from Fourier transforms between intensities according to the spatial van Cittert–Zernike [10, 11] theorem (see Sect. 4.6) and the related temporal Wiener–Khintchine [12] theorem, which play an important role in statistical optics [9, 13].

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4 Brightness and Coherence

The relations for a coherent source are given by the following general expressions Coherent source :

2  ∞ 2   −∞ |E(t)|  dt K = 

2  ∞ 

|E(0)|2   −∞ |E(t)|2  dt  2  ∞ 2   −∞ |E(r)|  dr H = 

2 .  ∞ 

|E(0)|2   −∞ |E(r)|2  dr

(4.24)

(4.25)

The corresponding expressions for a partially coherent source are Partially coherent source :  ∞  2 2  dt −∞ |E(t)|  ∞ K = 2

|E(0)|  −∞ |E(t)|2  dt  ∞  2 2  dr −∞ |E(r)|  ∞ . H= 2

|E(0)|  −∞ |E(r)|2  dr

(4.26)

(4.27)

The triangular brackets indicate a statistic average. If the fields are described classically with well-defined amplitudes, the triangular brackets become a cycle average. Using the above expressions, the spatial parameters H and h c and temporal parameters K and kc for Gaussian and flat-top source intensity distributions for both the coherent and chaotic cases are given in Table 4.1.

Table 4.1 Area–solid angle products H and time–energy products K for flat-top and Gaussian distributions that are either coherent or chaotic Coherent Partially coherent Distribution

H

hc

K

kc

H

Flat-top

1

1

1

1

1

Gaussian

1 4

1

1 2

1

1 2

hc

μ

Acoh As μ Acoh 2π(σrI )2

K 1 √1 2 = d 2 , or

kc

μ

τcoh τp μ τ √ coh I 2πσt

The spatial flat-top source distributions apply either for a box-like area, As a circular area As = π R 2 , and the effective Gaussian area is As = 2π(σrI )2 (see (4.8)). The micro coherence areas μ are denoted Acoh . The temporal effective pulse length for a flat-top distribution is τp = p /c and for √ a Gaussian distribution τp = 2π σtI (see (4.13) with τs = τp ). The microscopic coherence times μ are denoted τcoh

4.2 Brightness and Coherence

173

4.2.7 Photon Degeneracy Parameter The generalized peak brightness expression (4.23) for a coherent source, h c = kc = 1, is seen to be directly proportional to the coherent photon density, and it determines the degeneracy parameter according to n coh =

Bpeak H K λ3 . c   

(4.28)

Vcoh

According to Table 4.1 the coherence volume differs for flat-top (box volume) and Gaussian distributions according to Flat-top: Vcoh = λ3 , Gaussian : Vcoh =

λ3 . 8

(4.29)

The number of coherent photons per unit volume is linked to the brightness by (4.28). In this expression we have written the spatial and temporal uncertainties expressed by (4.19) and (4.20) in terms of the birth or coherence volume Vcoh which corresponds to setting the solid angle of emission and spectral bandwidth to unity, i.e. d = ω/ω = 1. Note that only the space-angle and time-frequency products are preserved for a given distribution shape. One may therefore define the birth volumes differently which lead to a concomitant change of the solid emission angle and bandwidth, since their product is conserved. In Fig. 2.23, we have defined the coherent birth area as λ2 /4, while one may also chose the value λ2 /π [14] which corresponds to the maximum resonant atomic interaction cross section with emission of a spherical wave, derived by Breit and Wigner [15, 16]. The maximum achievable brightness determines the maximum degeneracy parameter and reflects complete classical coherence. For a given wavelength, coherence can therefore be simply defined in terms of photon density or photon energy density instead of the conventional concept of wave interference. The equivalence of coherence and photon energy density is a consequence of the quantization of the EM field. In quantum mechanics, the classical concept of a phase relation between waves does not exist and is replaced by the concept of photon density (see Sect. 3.5).

4.2.8 Brightness of Storage Rings and XFELs As an example, let us link the degeneracy parameter of a state-of-the art XFEL with its peak brightness. We assume that the XFEL is operated in a low charge mode

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4 Brightness and Coherence

of 108 electrons (0.016 nC) per bunch that produces a coherent pulse of order of 1 fs length containing about 1010 photons, as previously shown in Fig. 1.11. Since the photons are coherent, the degeneracy parameter is also 1010 (photons). These photons are born in a Gaussian coherence volume Vcoh = λ3 /8 and are emitted into a unit solid angle, d = 1, with unit or 100% bandwidth (BW), ω/ω = 1. We are interested in the corresponding peak brightness in the conventional units [photons/s/mm2 /rad2 /0.1%BW] at a wavelength of λ = 0.1 nm (12.4 keV). The Gaussian coherence volume is then Vcoh = λ3 /8 = 1.25 × 10−22 mm3 , and the solid angle d = 1 sr corresponds to the product of two equal orthogonal planar half angles θ with units [rad], measured from the cone axis, that span the same cone, given in the small angle approximation (see footnote 3 and caption of Fig. 4.2 where θ = ϑ/2) by d  π θ 2 . The unit conversion in the brightness expression is thus 1 [sr] = π [rad2 ]. The 100% BW is simply reduced by 103 to obtain the proper brightness units. Inserting into the peak brightness expression (4.5) we obtain with c = 3 × 1011 mm/s, n coh c Vcoh dcoh ω/ω 1010 ∗ 3 × 1011 photons = 1.25 × 10−22 ∗ π × 106 ∗ 103 s mm2 mrad2 0.1%bandwidth photons . (4.30) = 0.76 × 1034 2 s mm mrad2 0.1%bandwidth

Bpeak =

Using the correlation of brightness and degeneracy parameter expressed by (4.30) we have plotted in Fig. 4.5 the average and peak brightness numbers for synchrotron radiation sources and the peak brightness for an XFEL source. On the right (black scale) we show the corresponding degeneracy parameter. For comparison we have also indicated the values for sunlight (n coh  3 × 10−4 ) and a common He:Ne laser (1 milliwatt power, n coh  3 × 109 ) at a wavelength of 633 nm [13]. Figure 4.5 illustrates that before the advent of XFELs, x-ray beams contained on average less than one photon per coherence volume, especially since beam line losses are not included in our numbers. It was therefore justified to neglect all interaction processes with matter where two or more coherent photons worked together. X-ray interactions occurred one-photon-at-a-time! The emitted intensity is typically stated in different units for synchrotron radiation, where it is specified as [number of photons of energy ω per time per area], and XFELs, where the intensity is given in units adopted from the laser community of [mJ/cm2 /fs]. Let us compare how the photon numbers in modern synchrotron experiments convert into XFEL intensities. To do so, we assume a photon energy of 1000 eV and assume that a third generation synchrotron undulator source produces about 106 photons of 10 eV (1%) bandwidth in a 100 ps long pulse that may be focused without loss to a 10 × 10 µm2 spot. With the conversion 1 eV= 1.6 × 10−16 mJ this gives an intensity of 1.6 × 10−6 mJ/cm2 /fs. This number compares to a typical XFEL intensity of order 103 mJ/cm2 /fs [18] for a comparable photon energy and bandwidth. The XFEL intensity is thus larger by a factor of about 109 , which roughly agrees with Fig. 4.5.

1 gen. X-FELs

2

30 25

10

5

2

2

He:Ne 10

10

ultimate rings 25 20

rd

3 gen. rings 20

nd

2 gen. rings

15 st

1 gen. rings 10 X-ray tubes 5 1900

1960

1980 Year

2000

1

10-5

10-10

Photon degeneracy parameter for =0.1nm

35 st

Log Peak Brightness (photons / s / mm / mrad / 0.1% BW)

175

2

Log Average Brightness (photons / s / mm / mrad / 0.1% BW)

4.2 Brightness and Coherence

sun

10-15

2020

Fig. 4.5 Historical and projected future increase in average brightness of storage rings (blue) and peak brightness of XFELs (red) [17]. On the right we have also given in black the number of photons per coherence volume or degeneracy parameter, calculated by assuming a source of wavelength λ = 0.1 nm. Note that, in practice, the achievable degeneracy parameter is lowered by beam line losses. On the right we indicate the degeneracy parameter of sunlight and a He:Ne laser at a wavelength of 633 nm [13]

4.2.9 Summary The brightness of a source is a fundamental measure of its quality. For synchrotron radiation sources, one typically states the average brightness while for XFEL sources the peak brightness. A source is diffraction limited if the quantity As d has the minimum value allowed by the uncertainty principle. A source is transform limited when the product τs ω/ω has the minimum value. All photons emitted by a diffraction and transform-limited source are coherent. They are born within a minimum volume of order λ3 and upon propagation remain contained in the coherence volume whose lateral area expands by coh . For a partially coherent or incoherent source, the number of photons in the coherence volume, called the degeneracy parameter, is smaller than the total number of emitted photons.

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4 Brightness and Coherence

4.3 Historical Descriptions of Coherence Historically, the concept of “coherence” has been used to break down the infinite complexity of light into some form of comprehensible order. This picture has evolved over time and so has our view of the nature of light. In practice, the description of the complex nature of light requires simplifying assumptions. The most important is space-time separability which has its origin in the observed transverse nature of light. It allows the separate discussion of the transverse or spatial properties of light in terms of spatial coherence and of the longitudinal or temporal properties of light in terms of temporal coherence. The convenient concept of space-time separability is bridged by two fundamental constants that characterize light and more generally the electro magnetic world. They are the constant speed of light c with dimension [distance/time], emerging from Einstein’s theory of special relativity [19], and Planck’s constant h with dimension [energy × time], discovered in the process of explaining the black-body spectrum [20, 21]. As shown by the dimension of c and h, the common parameter “time” links the classical concept of wavelength λ and the quantum mechanical concept of photon energy hc/λ.5 The practical importance of the concepts of spatial and temporal coherence is that they provide the foundation for understanding the observed behavior of light through diffraction and spectroscopy experiments. We can state as follows. The concept of “coherence” provides a way to describe the observed phenomenon of light diffraction. In practice, we assume space-time separability, leading to the concepts of spatial and temporal coherence as the foundation for understanding the measured behavior of light through diffraction and spectroscopy experiments. The two fundamental constants c [distance/time] and h [energy×time] provide the link between the classical concept of “length”, expressed by λ, and the quantum mechanical concept of “photon energy”, given by hc/λ. The concept of space-time separability greatly facilitates the development of a mathematical description of coherence. Below we will start by briefly outlining the geometrical concept of coherence dating back to Huygens, expressed through the magical Fresnel–Huygens principle. In its simplest form, coherence is related to the ability of waves to be in or out of phase, explaining the phenomenon of interference and diffraction. In Sect. 4.4 we will then introduce the more powerful formulation of coherence through the Fourier transform method. It is the most common way to link time and 5

In quantum mechanics, time is a parameter and is not expressed as the expectation value of an operator. This leads to a fundamental difference between the space-momentum uncertainty principle and the often used concept of a time-frequency uncertainty relation whose origin is somewhat problematic in quantum mechanics [15].

4.3 Historical Descriptions of Coherence

177

Fig. 4.6 Illustration of the length over which two waves that differ in wavelength by λ acquire a relative phase shift of λ/2

distance for which the phase shift is /2: =

2

energy in 1D and area and angle in 2D which we have implicitly used earlier to define the brightness of light. The third step in our discourse is the topic of Sect. 4.5. It involves an advanced formulation of “coherence” by including fluctuations of the measured light intensity, described by statistical optics through the spatiotemporal correlations between fields. In quantum mechanics, the fields are replaced by photon probability amplitudes under the assumption that photons are independent and non-interacting. The formulations of “coherence” emerging from statistical optics and quantum mechanics are equivalent and provide the link of the seemingly different concepts of coherence and brightness discussed earlier. Our final step requires a whole new chapter that will follow in Chap. 5. It discusses coherence within the complete framework of QED. It builds on the quantum mechanical formulation discussed in Sect. 4.5, which corresponds to the first order in QED, through an infinite series of increasing order. In QED, higher order coherence is manifested in correlations between an increasing number of photons. In the second order, corresponding to the correlation of two photons, the wave-particle duality already breaks down and we will specifically explore this all-important transition.

4.3.1 Geometrical Coherence Concepts in Time and Space Classical coherence assumes a continuous light wave of wavelength λ with oscillation time τ0 = λ/c or cycle frequency ν = 1/τ0 that has no beginning and no end. Temporal coherence may be defined6 by considering when two such waves of slightly different wavelengths launched in-phase at the same point become out of phase by λ/2 as illustrated in Fig. 4.6.

The simple geometrical coherence condition λ = λ/2 used here [22] is different from the condition λ = λ/(2π ) used in the Fraunhofer approximation and that obtained from the Fourier transform for a coherent flat-top field distribution, λ = λ, but is the same as for a coherent Gaussian distribution (see Table 4.1).

6

178

4 Brightness and Coherence Geometrical optics coherence cone and area (a) point source

d point source

coh

=

(b) finite-sized source distant plane

Bcoh z02

d d

coh

z0

S

z0 -
1 for paramagnetic but < 1 for diamagnetic media and in such systems typically differs from unity by a factor of about 10−5 . In general, μ and μr of a medium are complex and second rank tensors. We shall only consider isotropic media where they are scalars. In order to describe the interaction of the rapidly changing H-field of the EM wave with the magnetization of a material we have to consider the frequency dependence of M. This requires treatment of the permeability as a complex quantity as discussed in more detail in Sect. 7.3 below, where we shall indicate the complex nature by a tilde and we have μ˜ = μ˜ r μ0 . At low frequencies, μ˜ r = μr is real to a good approximation and in ferromagnetic materials μr is typically much larger than 1. Mu-metal (approximately 75% nickel, 15% iron, plus copper and molybdenum) can have relative permeabilities of 80,000–100,000 which makes it very effective at screening static or low-frequency magnetic fields. The relation between the magnetization M and the magnetic field H defines the often quoted magnetic susceptibility χm . Combining (7.1) and (7.2) we obtain M = (μr − 1)μ0 H = χm μ0 H.

(7.3)

The magnetic susceptibility χm = μr − 1 is dimensionless. For paramagnetic and ferromagnetic media we have χm > 0 and for diamagnetic media χm < 0. Analogous equations exist for the electric field vector E and the dielectric field or dielectric displacement D. In vacuum and in many materials of interest we have D = 0 E, where 0 = 8.86 × 10−12 A s V−1 m−1 is the dielectric constant or electric permittivity of the vacuum. To generally describe the electric fields in matter, the dielectric polarization P is introduced which is the density of electric dipoles. The three electric vectors are related by the equation D = 0 E + P,

(7.4)

D and P have units [ A s m−2 ], and 0 = 8.86 × 10−12 A s V−1 m−1 is the dielectric constant or permittivity of the vacuum. A material with a quasi-permanent electric charge or electric polarization P is called an electret, a name coined by Oliver Heaviside in 1885 from “electricity” and “magnet”. In air, electrets attract stray charges from the surroundings which soon neutralize the surface charges. Some materials also display ferroelectricity, i.e. they exhibit external electric field hysteresis. Similar to the relationship (7.2) for the magnetic vectors we have

320

7 Semi-classical Response of Solids to Electromagnetic Fields

D =  E = r 0 E.

(7.5)

Here  is the absolute electric permittivity and has the same units as the electric permittivity of free space 0 , while the relative electric permittivity r is dimensionless. Like the relative permeability μr , in general, the relative permittivity r is complex and a tensor. We shall only consider isotropic substances, but in order to account for its frequency dependence we shall in Sect. 7.3 below consider the complex nature of r , indicated as ˜r , also called the complex dielectric function, The electric susceptibility χe is defined in analogy to (7.3) between the dielectric polarization P and the electric field E according to P = (r − 1)0 E = χe 0 E.

(7.6)

The electric susceptibility χe = r − 1 is dimensionless. In contrast to the magnetic field vectors, the electric field vectors do not indicate a sense of rotation, they only define a direction and do not change sign when the time is inverted. The electric field vectors are polar vectors.

7.3 Frequency Response of Materials: Microwaves to X-Rays We consider a material exhibiting both a static magnetic polarization M given by (7.3) and dielectric polarization P given by (7.6) and then consider its response with increasing frequency.

7.3.1 Permittivity: Electric Field Response According to Maxwell’s equations, time-varying magnetic fields are always accompanied by electric fields. Classically, a material’s ability to prevent field penetration by internal screening is described for the electric fields by the complex permittivity and for the magnetic fields by the complex permeability. The complex frequency dependent dimensionless relative permittivity, ˜r , is usually written as2 ˜r (ω) = 1 (ω) + i 2 (ω) .

(7.7)

The real part 1 is related to the stored energy within the medium, and the imaginary part 2 is related to the dissipation of energy within the medium. For metals, key parameters that determine the frequency dependence of ˜r (ω) are the relaxation time, 2

We have used the phase convention that the time evolution of the electromagnetic wave is given by exp(−iωt).

7.3 Frequency Response of Materials: Microwaves to X-Rays

321

τ E , of the conduction electrons to the E-field excitation and the plasma frequency ω P . In metals, electromagnetic waves with frequencies below ω P are strongly absorbed because the electrons are able to respond fast enough to screen the incident electric field. They are then re-emitted, and the reflectivity is near unity. For waves with frequencies above ω P the electrons are no longer able to follow and screen the electric field, and the waves can penetrate into the metal. In most metals, the plasma frequency is in the ultraviolet (several eV), making them shiny (reflective) in the visible range. At frequencies ω  ω P , one can describe ˜r (ω) by the following expressions for the real and imaginary parts3 1 (ω) = 1 −

ω2P τ E2 ω2P τ E , and  = 2 1 + ω2 τ E2 ω(1 + ω2 τ E2 )

(7.9)

where τ E (units [s]) is the conduction electron relaxation time in response to the E-field and ω P is the plasma frequency (units [rad/s]). When evaluating (7.9), one needs to be careful because all frequencies ω are angular frequencies (units [rad/s]) which are related to the cycle frequencies ν (units of [Hz] ≡ [s−1 ]) by ω = 2π ν. Typically, one plots the permittivity as a function of the cycle frequency so that one needs to write ω P = 2π ν P and ω = 2π ν in (7.9) with the frequencies ν P and ν in units of [Hz]. The real and imaginary parts obey the Kramers-Kronig relation (see Sect. 6.6), and we have plotted the frequency dependent quantities 1 and 2 for Fe metal in Fig. 7.1. In the figure we have indicated the characteristic E-field induced relaxation rate of the electrons where −1 = 2 . For Fe with τ E = 34 fs [3] we have 1/2π τ E = 4.6 THz. The gray shading indicates the region where the response is dominated by the imaginary part |2 |.

7.3.2 Permeability: Magnetic Field Response Similar to the E-field response inside a material, which is expressed by the complex relative permittivity, the H-field response is characterized by the dimensionless complex relative permeability. It is written as μ˜ r (ω) = μ1 (ω) + i μ2 (ω),

(7.10)

At frequencies below about 1 THz, 1 and 2 are also related to the components of the complex Drude conductivity given by   ωτ 1 + i (7.8) , σ˜ (ω) = σ1 (ω) + i σ2 (ω) = σ0 1 + ω2 τ 2 1 + ω2 τ 2

3

where σ0 is the DC electrical conductivity and τ the Drude relaxation time, usually assumed to be τ = τE .

7 Semi-classical Response of Solids to Electromagnetic Fields

Real & imag. parts of ~r and ~ r

322

109

Fe metal microwave response

107

1 /2

5

103

E

-

10

1 /2

H

101

10-1 10-1

1

10

100

10 3

4

10

Frequency (GHz)

Fig. 7.1 Frequency dependence of the real and imaginary parts of the dimensionless relative permittivity, 1 and 2 , and relative permeability, μ1 and μ2 , for ferromagnetic Fe metal. The red circles and black diamonds are derived from experimental data by Ordal et al. [3, 4]. The permittivity curves were calculated according to (7.9) with the parameters ω P /2π = 885 THz and τ E = 34 fs [3]. The permeability curves were calculated according to (7.11) with the parameters μr0 = 100 and τ H = 100 ps. The gray shading indicates the region where the response is dominated by the imaginary part |2 |

and the ratio of the imaginary to the real part μ2 (ω)/μ1 (ω) of the complex permeability is called the loss tangent and provides a measure of how much power is lost in a material versus how much is stored. When losses occur, B lags behind H. The general frequency dependence of the permeability is quite complicated. Losses can be due to different mechanisms and are dependent on the composition, the microstructure and macroscopic shape of the magnetic material. Damping phenomena may arise from eddy currents, magnetic domain wall motions, and damping of the precession of the magnetization [5–7]. While the resonant behavior of μ˜ r or the related AC susceptibility χ˜ = μ˜ r − 1 can be probed in the GHz regime under application of an external magnetic field by ferromagnetic resonance, the general behavior of μ˜ r for electromagnetic fields up to the optical regime is seldom discussed because of its complexity. From experiment one knows two limits. Gradual alignment of the random magnetic domains of a sample by application of an H-field yields the real DC permeability μr0 , determined from the initial slope of the magnetization curve. For Fe, for example, one finds μr0  100. We take this real DC permeability μr0 as our constant low-frequency value for μ˜ r , with a negligibly small imaginary part. The other limit is the well-studied optical regime starting at photon energies around 1 eV or frequencies ≥ 250 THz. Here μ˜ r is also known to be real and has the simple real value 1, even for magnetic materials. That is why the H-field response is typically not even considered in electromagnetic response theories at optical and higher frequencies.

7.3 Frequency Response of Materials: Microwaves to X-Rays

323

By considering the intrinsic low-frequency response of a ferromagnetic material to H-fields one may write down a general response equation for the real and imaginary parts of μ˜ r . The key parameter is the H-field response time of the magnetic moments in the sample, assumed to be aligned in local domains that point into different directions. The H-field applies torque on the atomic moments and the characteristic response time τ H is that associated with the magnetic anisotropy energy which favors the original magnetic alignment in the domains. Typical relaxation times are τ H = 100 ps corresponding to a change of μ˜ r in the range between about 1 GHz and 1 THz. One may quite generally write μ˜ r (ω) in terms of the following Kramers-Kronig-related real and imaginary parts μ1 (ω) = 1 +

μr0 − 1 ω τ H (μr0 − 1) and μ (ω) = , 2 1 + ω2 τ H2 1 + ω2 τ H2

(7.11)

where μr0 is the real DC permeability and τ H the spin relaxation time in response to the H-field. The expected frequency dependence of the quantities 1 and 2 for μr0 = 100 and τ H = 100 ps, characteristic of Fe metal, is plotted in the lower half of Fig. 7.1, where we have also indicated by a vertical line at μ1 = μ2 the characteristic H-field induced relaxation rate of the spins 1/2π τ H = 1.6 GHz.

7.3.3 From Permittivity and Permeability to Optical Parameters Above the microwave regime, at frequencies exceeding about 1 THz, the propagation of electromagnetic (EM) fields into an isotropic magnetic medium is typically described by the complex refractive index, n, ˜ composed of a real part, the real refractive index, n, and an imaginary part, the extinction coefficient, κ, according to [8, 9] n˜ 2 (ω) = [n(ω) + i κ(ω)]2 =

˜ (ω) μ(ω) ˜ = ˜r (ω)μ˜ r (ω). 0 μ0

(7.12)

The quantities n and κ are usually referred to as the “optical constants”. Owing to their strong energy dependence, we will here refer to them instead as the optical parameters. They describe the photon energy dependent refraction and absorption of the fields, respectively. The master equation (7.12) links n˜ to the complex electric permittivity ˜ and magnetic permeability μ. ˜ From (7.12), we obtain the general expressions (7.13) n 2 − κ 2 = 1 μ1 − 2 μ2 , 2 n κ = 1 μ2 + 2 μ1 .

324

7 Semi-classical Response of Solids to Electromagnetic Fields

The real and imaginary parts of the refractive index are then given by  n= and

(1 μ1 − 2 μ2 )2 + (1 μ2 + 2 μ1 )2 + 1 μ1 − 2 μ2 2

(7.14)

 (1 μ1 − 2 μ2 )2 + (1 μ2 + 2 μ1 )2 − 1 μ1 + 2 μ2 . κ= 2

(7.15)

For non-magnetic materials we have μ1 = 1 and μ2 = 0 so that n 2 − κ 2 = 1 , 2 n κ =  2

(7.16)

and

Non-magnetic: n =

   2 + 2 +   1 1 2 2

, κ=

   2 + 2 −   1 1 2 2

.

(7.17)

The real part n of the complex refractive index, and the imaginary part κ, which characterizes extinction or absorption are plotted for Fe metal in Fig. 7.2. The limited available experimental data [3, 4] are shown as blue and red circles, while the solid blue and red curves are calculated by use of (7.14) and (7.15) with the permittivities and permeabilities plotted in Fig. 7.1. The solid black curves in Fig. 7.2 are calculated assuming non-magnetic Fe according to (7.17) using the permittivity values in Fig. 7.1. They merge into the blue and red magnetic curves above about 103 GHz =1 THz, revealing that the magnetic permeability ceases to be important. Since the permeability reflects the response of the H-field component of the EM wave, we see that in the THz region, the influence of the H-field ceases to be important and the EM field response can be phrased in terms of that of the E-field alone. At frequencies above about 1 THz, the complex refractive index n˜ and the optical parameters n and κ are determined by the electrical permittivity alone. The material response to an EM wave can then be phrased in terms of the E-field alone and the effect of the H-field can be neglected.

7.3 Frequency Response of Materials: Microwaves to X-Rays

325

Fe metal microwave response

Optical constants n and

105 104

n ma ongne tic

1/2

n

103 1/2

102

E

H

10 1 10-1

1

10

100

10 3

4

10

Frequency (GHz)

Fig. 7.2 Frequency dependence of the optical parameters n and κ. The blue and red circles are from [3, 4]. The solid red and blue curves are calculated by use of (7.14) and (7.15) with the permittivities and permeabilities in Fig. 7.1. The solid black curves are calculated assuming nonmagnetic Fe according to (7.17) with the permittivity values plotted in Fig. 7.1. Note that they merge into the magnetic curves above about 103 GHz. The gray shaded area is that shown in Fig. 7.1

7.3.4 Penetration Depth of EM Waves: From Microwaves to X-Rays We briefly want to follow up on the important point that above the microwave region (above 1 THz  4 meV) the interaction of EM waves with matter no longer depends on the H component of the EM wave and can be described through the frequency dependent response of the electrons to the E-field. The permeability then becomes unimportant, and the optical parameter description describes the dielectric response to the E-field. One also changes notation, renaming κ = β which describes the important absorptive response that determines the penetration depth into materials. Above the microwave regime, the characteristic penetration depth or absorption length x at a given wavelength λ is defined as the distance over which the intensity decreases by a factor 1/e or about 35% (see Sect. 7.4.2 below) as x =

λ . 4πβ

(7.18)

It is half of the so-called skin depth typically used in the microwave region. In order to get an overall perspective of the change of x , we have plotted it for Fe metal in Fig. 7.3 over the entire spectral range from microwaves to x-rays, spanning about ten orders of magnitude. Here the transition from the microwave region to higher frequencies is of particular interest. In the microwave region, the absorption length in Fe is dominated by the

326

7 Semi-classical Response of Solids to Electromagnetic Fields

6

10

Wavelength ( m) -2 2 1 10 10

3

10

-4

10

Photon energy (THz) -2

6

EM absorption length

1

10

2

10

4

6

10

Fe metal 5

10

X

(nm)

10

10

4

10

H-field response dominates

E-field response dominates

3

K

L

10

2

10

M

10 microwaves

1 -6 10

-4

10

infrared

VUV

-2 2 1 10 10 Photon energy (eV)

x-rays 4

10

Fig. 7.3 Characteristic 1/e absorption length x of electromagnetic radiation for Fe metal from microwaves to x-rays. The approximate name of the different spectral regions is indicated at the bottom. Below about 1 THz (top scale) the response of ferromagnetic Fe is dominated by the interaction of the EM H-field with the magnetization, shown as a red curve. The dashed black curve neglects the H-field response and considers only the E interaction with the electrons which completely dominates in the optical and x-ray regions (solid black curve)

interaction of the EM H-field with the magnetization, as shown by the red curve. The H-field contributes through the DC magnetic permeability (also see Fig. 7.1), resulting in enhanced absorption and reduced penetration length relative to the dashed black line which ignores magnetic effects. The important phenomenon may be understood in a simple physical picture as follows. A frequency of 1 THz corresponds to an EM cycle time of 1 ps. At subps cycle times, the H-field oscillations become so fast that the spins can no longer follow because the H-field induced torque on the spins requires transfer of angular momentum to the lattice [1]. This however becomes bottlenecked in the range below about 1 ps [10, 11]. At shorter EM cycle times only the electrons can respond and their response follows the E-field. In the following we will consider the behavior above the microwave range, above the red demarkation line in Fig. 7.3 and consider both the refractive and absorptive response determined by n(ω) and κ(ω) = β(ω).

7.3 Frequency Response of Materials: Microwaves to X-Rays

327

7.3.5 Optical Parameters: From the THz to the X-Ray Regime Above the microwave region, the dielectric sample response is then phrased in terms of the dimensionless complex refractive index n(ω), ˜ composed by its dispersive real part n and imaginary absorptive part κ = β (sometimes denoted n = n  and κ = n  ). In the optical region the response is described by (7.12), or Optics: n(ω) ˜ = n(ω) + i β(ω)

(7.19)

where n and β (n, β > 0) are the dimensionless energy dependent optical parameters. This formulation can be extended to the x-ray regime, and as an example we show in Fig. 7.4 the characteristic change of n and β from the THz to the x-ray regime for Fe metal. The figure shows that at photon energies above about 10 eV, n(ω) approaches unity and in fact becomes slightly less than 1 for Fe metal. This has the important consequence that below a certain “critical” grazing incidence angle x-rays are totally reflected. In the ultraviolet and x-ray regions one conveniently uses the following

10

2

1

3

Photon energy (cm -1) 4 5 6 10 10 10 Photon energy (THz) 2 3 4 10 10 10

10 10 2

10

10

10

Wavelength ( m) 10-1 1

10

10 10-2

7

5

10

10 10-3

8

6

10-4

=

2

Fe metal Optical constants n and

10

n

1 10 10 10 10 10

M-edge

-1

=1- n -2

10

-2

-3

10

-3

-4 -5

10

-5

100

10

Kedge

-4

10

L-edge

Ledge

-6

0.01

1000 Photon energy (eV)

0.1

4

10

1 10 100 Photon energy (eV)

K-edge

1000

10 4

Fig. 7.4 Optical parameters n and β for Fe metal derived from experimental data [1, 12, 13]. Due to the large energy range we show various energy scales and also the wavelength scale in μm. Near the L-edge we have included the pronounced resonant contributions as discussed below. Near the L-edge we have included the resonant contributions (see Fig. 7.12). In the inset we show the energy dependent refractive index δ = 1 − n, defined in (7.20)

7 Semi-classical Response of Solids to Electromagnetic Fields

Fig. 7.5 a Comparison of the magnitudes of δ = 1 − n and β for Fe metal, derived from the data in Fig. 7.4 above 100 eV. Note that δ becomes negative right before the L-edge position (not shown). b Scaling of δ and β with wavelength λ, which at higher photon energies increasingly becomes β ∝ λ4 and δ ∝ λ2

-1

10

Fe

-2

10

(a)

-3

10

-4

10

-5

10

-6

10

Optical constants

328

Kedge

Ledge

-7

10

-8

10 -2 10

(b) 2 -3

10

3.8 -4

10

-5

10

-6

10

100

1000

4

10

Photon energy (eV)

terminology [9, 14] X-rays: n(ω) ˜ = 1 − δ(ω) + i β(ω),

(7.20)

where the real part is redefined as n = 1 − δ. The choice of the positive sign of the imaginary part in (7.19) and (7.20) is consistent with our phase choice for propagating electromagnetic waves, given by (3.11). According to Fig. 7.4, β ≥ 0 decreases strongly with photon energy, leading to greater penetration of x-ray than optical radiation into materials as shown in Fig. 7.3. The replacement n = 1 − δ is reasonable when |δ|  1, so it can be regarded as a correction. A direct comparison of the sizes of β and δ for Fe is shown Fig. 7.5a in the energy region > 100 eV where n is close to unity. Outside the resonance regions, δ(ω) is positive, but at the L3 absorption resonance it is negative as shown in more detail in Fig. 7.12. Both β and δ decrease with photon energy, and above about 1000 eV the absorptive index β typically becomes smaller than the dispersive index δ. In the hard x-ray region they scale as β ∝ λ4 and δ ∝ λ2 , as illustrated in Fig. 7.5b.

7.3 Frequency Response of Materials: Microwaves to X-Rays 1

(a)

Polystyrene

0.006

329

(b)

C K-edge region

0.004

0.5

and

0.002

0 0

-0.5

-0.002

-1 1

-0.004

10

-0.006

100 275 Photon energy (eV)

280

285

290

295

300

Fig. 7.6 a Energy dependence of the optical parameters β and δ for polystyrene over a wide photon energy range [17]. b Enlarged region near the C K-edge (see Footnote 4)

Another example of the energy dependence of δ and β from the optical to the x-ray range is shown for polystyrene (PS) in Fig. 7.6a, and in (b) we show the details around the C K-edge region.4 In PS, β is about zero in the optical region since no optical transitions are possible across the large band gap of about 4.2 eV. Transitions from the highest occupied (HOMO) to the lowest unoccupied (LUMO) molecular orbitals become visible in the 4–10 eV region. Both δ and β become small and smooth above about 100 eV, until the carbon K-shell excitation channel opens up around 285 eV. The rich fine structure in δ and β near the C K-shell absorption threshold is shown enlarged in Fig. 7.6b. It results from transition from the K-shell to empty molecular orbitals [18]. In Fig. 7.7 we show the values of β and δ for a hydrocarbon of composition C8 H8 , which is the same as PS. The plot is arranged similar to that for Fe in Fig. 7.5 and also shows the scaling of the β and δ with wavelength λ. It shows that at a photon energy of 10 keV δ is about a factor of 103 larger than β. This has important consequences for x-ray protein crystallography, which utilizes x-rays around 10 keV. The absorptive (β) response, which results in the creation of photo- and Auger electrons that cause deleterious radiation damage, is greatly reduced relative to the dispersive (δ) response which enters on an equal footing as non-resonant Thomson (∝ r0 Z ) scattering. It also makes it advantageous to utilize phase contrast imaging (δ) [19] rather than absorption contrast (β) imaging, typically used in the soft x-ray region [20]. We can summarize as follows. The effective medium response of matter to x-rays, described by the semiempirical and phenomenological optical parameters, simplifies in the x-ray

4

The data shown in Fig. 7.6b were measured with somewhat lower photon energy resolution than those published in [15] (also see Suppl. Mat. in [16]).

-1

10

-2

10

(a)

Hydrocarbon C8H8

-3

10

-4

10

-5

10

-6

10

-7

and

Fig. 7.7 a Comparison of the magnitudes of δ = 1 − n and β for a hydrocarbon above 100 eV. We have ignored the detailed resonant contributions near the C K-edge which depends on C-C bond hybridization. b Scaling of δ and β with wavelength λ, which in the x-ray region is well approximated by β ∝ λ4 and δ ∝ λ2

7 Semi-classical Response of Solids to Electromagnetic Fields

10

-8

10

-9

Optical constants

330

10 -3 10

10

-4

10

-5

10

-6

10

-7

(b)

2

4

100

1000

4

10

Photon energy (eV)

relative to the optical regime. With increasing photon energy, the absorptive response, characterized by β, decreases and at energies around 1 keV become weaker than the dispersive response, expressed by δ. In the hard x-ray region, the response scales with wavelength as β ∝ λ4 and δ ∝ λ2 .

7.4 Dielectric Response Formulation of X-Ray Absorption Here we derive three equivalent formulations for the exponential x-ray absorption law based on the phenomenological effective medium response, introduced above as an extension of the low-frequency response.

7.4.1 Optical Parameter Formulation X-ray absorption corresponds to the attenuation of electric fields transmitted through a sample whose electronic response is expressed by the complex refractive index

7.4 Dielectric Response Formulation of X-Ray Absorption

331

written in the form of (7.20). We assume that the incident field propagates in vacuum in the z direction and traverses a sample in the form of a film in the x-y plane of thickness d along z. For simplicity we shall ignore the time dependence of the field and consider only the change of the field as a function of distance z in the sample. The incident field is then given by

 Ein (z) = E0 ei kz + e−i kz = 2 E0 cos(kz).

(7.21)

The first term in the square brackets causes electronic excitations in the sample, while the second term causes stimulated de-excitations from the excited state. Since we assume that all atoms in the sample are originally in their ground state, the second term is not active and can be omitted. The starting point for our description is therefore an incident field of the form (7.22) Ein = E0 eikz . The attenuation of the field upon transmission through a sample is illustrated in Fig. 7.8. We define the entry point of the electric field in the sample as z = 0 and assume a non-magnetic sample so that the refractive index is determined by the response of the charge in the sample, only. After having traversed a distance z in the sample, the field is given by ˜ −βkz = E0 eikz e−iδkz (7.23) E = E0 einkz

  e     phase shift

Ein

reduction

so that for a sample thickness d the field emerging from the sample, i.e. the transmitted field, is given by −βkd Etrans = Ein e−iδkd

  . e   . dispersion

(7.24)

absorption

where we have indicated the dispersive contribution e−iδkd = e−i2πδd/λ , leading to a phase shift φ = −2π δd/λ in the transmitted relative to the incident field, and

n =1

Ein = E0 eikz Filed amplitude or Intensity

Fig. 7.8 Decay of the amplitude E and intensity I ∝ |E|2 of an incident wave E in = E 0 eikz during transmission through a homogeneous sample with complex index of refraction n˜ = 1 − δ + iβ

~

n =1

+i ~

E0 e inkz

E = Ein e

i kz

I = Iin e z

2 kz

kz

332

7 Semi-classical Response of Solids to Electromagnetic Fields

the real absorptive contribution which reduces the incident amplitude E 0 by a factor e−βkd = e−2πβd/λ . The dispersive phase shift in the transmitted field arises from the electronic fields created inside the sample in response to the incident field. The incident field causes a charge oscillation in the material, and the oscillating dipoles in turn generate a (scattered) field that is phase shifted and contributes to the total transmitted field. The real exponential causes a decrease of the incident field amplitude E in in the material determined by the absorptive coefficient β. By forming the absolute value squared of the transmitted field (7.24) we obtain the law for the transmitted intensity, defined by (3.120) as I = 20 c |E|2 . Itrans = I0 e−2β k z

(7.25)

7.4.2 Absorption Coefficient Formulation The absorption law given by (7.25) may also be derived by defining a characteristic length x in a material over which the transmitted intensity decreases by a factor 1/e or about 35%, corresponding to 2βkz = 1 in (7.25). One then calls the quantity μx =

1 4πβ = 2kβ = x λ

(7.26)

with the dimension [1/length] the linear absorption coefficient. In this formulation the intensity is attenuated by the amount μx dz as it traverses the thin sheet of thickness dz at a depth z from the surface so that −

dI (z) = μx I (z). dz

(7.27)

The differential equation has the solution I (z) = A e−μx z and by choosing A = I0 as the incident intensity at the point z = 0 we obtain Itrans = I0 e−μx z

(7.28)

By use of (7.26) this is seen to be an equivalent formulation of (7.25). The energy dependence of the absorption coefficient μ has previously been shown for water in Fig. 1.16 and for Co metal and CoO in Fig. 1.17. In the insulator CoO, for example, absorption is absent in the optical range due to the existence of a 2.5 eV band gap. A similar response is observed in Fig. 7.9 for the absorption coefficient μ of polystyrene, whose optical parameters were previously shown in Fig. 7.6.

7.4 Dielectric Response Formulation of X-Ray Absorption

333

-1

10

-2

Polystyrene

C K-edge fine structure C K-edge

Absorption coefficient

x

(nm -1 )

0.02

10

0.01

10

-3

10

-4

band gap ~ 4.2 eV

1

0

10

100

280 Photon energy (eV)

290

300

310

Fig. 7.9 Energy dependence of the absorption coefficient μx for polystyrene from the optical to the x-ray region and near the C K-edge [15, 17]. Also see Fig. 7.6 for the corresponding optical parameters

7.4.3 Beer-Lambert Formulation We finally discuss another equivalent formulation of the exponential absorption law expressed by (7.25) and (7.28). It is referred to as the Beer-Lambert law in honor of the two scientists who first suggested it and reads5 Itrans = I0 e−σabs ρa z .

(7.29)

It is expressed in terms of the atomic absorption cross section σabs and the atomic number density ρa . We previously linked the cross section σabs to the imaginary part of the atomic scattering factor by (6.72) as σabs = 2λr0 f 2 = 2λ f  =

4π β. λρa

(7.30)

We will formally establish the relation between f 2 (and f  ) and β in the next sections below which will prove the equivalence of (7.25) and (7.29). The atomic number density ρa is calculated from the tabulated atomic mass density, ρm [mass/volume], the atomic mass number, NM [mass/mol], and Avogadro’s number NA = 6.02214 × 1023 [atoms/mol] according to ρm NA ρa = . (7.31) NM

5

Historically, Lambert found that the absorbance is proportional to the thickness d of the sample, and Beer inferred that the absorbance is proportional to the concentration of the sample (ρa ).

334

7 Semi-classical Response of Solids to Electromagnetic Fields

In summary, the absorption law for a sample of thickness d can therefore be written in the equivalent forms Itrans = I0 e−2β k d = I0 e−μx d = I0 e−σabs ρa d = I0 e−2λ r0

f 2 ρa d

.

(7.32)

7.5 From Dielectric to Atom-Based X-Ray Response In the following sections we will link the description of x-ray interactions with matter, framed above in the framework of dielectric response through the phenomenological optical parameters, to an atom-based formalism based on the x-ray scattering factors introduced in the previous Chap. 6. The link between the two formulations is suggested by the formal similarity of the expressions for the total atomic scattering factor expressed by the Q independent form of (6.29) F(ω) = Z + F  (ω) − i F  (ω)

(7.33)

and the expression for the refractive index (7.20), or n(ω) ˜ = 1 − δ(ω) + i β(ω).

(7.34)

One might therefore expect that the energy dependent real terms Z + F  (ω) and 1 − δ(ω) and the imaginary terms F  (ω) and β(ω) can be related. In order to do so, we first need to extend the formulation of the atomic response in terms of the scattering factors, formulated for single atoms in the last chapter, to the collective response of many atoms. We will accomplish this through an atom-byatom building block description of the x-ray response, starting from single atoms to a thin sheet and then to a thick film or a solid. We first briefly revisit the atom-based formalism of scattering introduced in Chap. 6.

7.5.1 Brief Review of X-Ray Scattering Factors The complex atomic scattering factor given in units of [ number of electrons per atom] is defined by (6.29) or F(Q, ω) = F 0 (Q) + F  (ω) − i F  (ω).

(7.35)

Here the energy independent atomic form factor F 0 (Q) describes the angle or Qdependence of the x-ray interaction with the entire atomic charge density. The two

7.5 From Dielectric to Atom-Based X-Ray Response

335

energy dependent “anomalous” scattering factors F  (ω) and F  (ω) describe the strong energy dependent deviation of scattering near absorption edges.6 The related complex atomic scattering length of dimension [length] is obtained by multiplying the atomic scattering factor with the Thomson scattering length r0 = e2 /4π 0 m e c2 = 2.818 × 10−15 m/electron, and accounting for the polarization dependence of scattering by a factor  ·   , so that f (Q, ω) = f 0 (Q) + f  (ω) − i f  (ω)

 = ( ·   ) r0 F 0 (Q) + F  (ω) − i F  (ω) .

(7.36)

In the soft x-ray range or for scattering in the forward direction, where the momentum transfer Q is small, we have f 0 (Q → 0) = Z , where Z is the number of electrons per atom. This allows one to rewrite (7.35) in terms of the web-tabulated Henke-Gullikson factors [13, 21] f 1 (ω) and f 2 (ω) according to (6.67) or7 F(ω) = Z + F  (ω) − i F  (ω) .     f 1 (ω)

(7.37)

f 2 (ω)

The corresponding atomic scattering lengths are then   f a (ω) = (  · ) r0 Z + f  (ω) −i f  (ω) .

    r0 f 1 (ω)

(7.38)

r0 f 2 (ω)

7.5.2 From Single Atoms to Atomic Sheets We start our building block approach of atomic response by considering a thin atomic sheet, as illustrated in Fig. 7.10. We express the transmission of the field through the sheet through atomic scattering instead of the dielectric response in Sect. 7.4. As shown in (a) we assume an incident EM wave of wavelength λ that is larger than the thickness  of the atomic sheet so that the interaction may be described in the dipole approximation. We assume that the incident wave has a planar wavefront across a homogeneously illuminated sheet area, defined by a circular aperture of area A = π R 2 , that contains a number of atoms Na defined by the atomic number density ρa = Na /(A). The sheet is thus coherently illuminated. We now consider elastic scattering of the atoms in the sheet by use of the first Born approximation discussed in Sect. 6.5, and are interested in the scattered field at a point O on the optical axis, at a large distance z 0 from the sheet. The scattered Born 6 The term “anomalous” has been maintained today in the important “multiple-wavelength anomalous diffraction” (MAD) technique in x-ray crystallography as discussed in Sect. 8.11. 7 Unfortunately, the historical notation is somewhat confusing because the scattering factors f and 1 f 2 and the scattering lengths f  and f  in (7.36) are both denoted by a small f .

336

7 Semi-classical Response of Solids to Electromagnetic Fields

Fig. 7.10 a X-rays of wavelength λ traverse a thin atomic sheet of thickness . b The x-rays are described by a plane wave E in = E 0 exp(ikz) incident at normal incidence on an area A = π R 2 of the thin sheet containing Na atoms and we are interested in the field E scat at the point O on the optical axis at a distance z 0 . c Definition of key distances used for the derivation of the field E scat

(a)

= 0.25 nm

= 1.6 nm

z

k atoms

(b)

A= R 2

z0 Na atoms

E0 e ikz

k

Escat

sample plane with atomic scatterers

(c)

P r

optical axis

l z0

O

field consists of contributions from the Na atoms in the illuminated sheet area A and we have to integrate over the contributions from all points P in the sample plane,

separated from our observation point O by distances  = z 02 + r 2 as illustrated in Fig. 7.10c. Since the fields are forward scattered with Q  0, we can denote the atomic scattering length as f (Q) = f a and obtain the scattered field at point O by integrating in cylindrical coordinates as

O E scat

R ik √z02 +r 2 Na e = −E 0 f a 2π r dr . 2 πR 2 2 z + r 0 0

(7.39)

We obtain Na f a λ ikz0 Na f a λ ik √ R 2 +z02 O = −iE 0 e + iE e . E scat 0 2 π R 2  π R   sheet scattering

(7.40)

edge diffraction

As indicated by underbrackets, the on-axis scattered field arises from the contributions of all atoms in the sheet, plus a contribution from the atoms at the perimeter of the aperture at distance R. The perimeter atoms cause an edge diffraction effect when the fields are calculated off-axis, leading to the well-known Airy rings. The

7.5 From Dielectric to Atom-Based X-Ray Response

337

edge diffraction contribution disappears with increasing R due to the increasingly smaller fraction of perimeter atoms.8 The forward scattered field by the sheet (without edge diffraction) is seen to be proportional to the number of atoms so that the scattered intensity scales with Na2 , revealing the coherent superposition of the fields due to the cylindrical symmetry about the optical axis. The total on-axis Born field, given by (6.49), is obtained by adding the incident field to the scattered field, and we obtain the following total field at point O O = E 0 eikz0 − i E trans   hole E trans

Na f a λ E 0 eikz0 . A  

(7.42)

sheet E scat

The new label “trans” indicates that the field at O can also be taken to represent the transmitted field in the forward direction. Like the scattered field from a single atom, it contains an unscattered contribution from the incident field transmitted through the “hole” (no film) plus a scattered contribution from the atoms in the film.

7.5.3 Atomic Scattering Factors: Born Approximation Versus Huygens–Fresnel Principle The magical Huygens–Fresnel (HF) principle allows the calculation of the diffraction pattern of an aperture whose open area is vacuum so that there are no true scattering centers within the aperture hole. Historically, this caused little concern because light propagation was envisioned to require a medium, the infamous “aether”, so that an aperture could be imagined as being filled by the aether medium. In modern physics, the idea that a point scatterer converts a plane wave into spherical waves is the essence of the first Born approximation discussed in Sect. 6.5. In the x-ray region, where scattering is due to atoms, one may thus compare the Born scattering of real atoms with the HF scattering of virtual atoms (or points) of the same number density. In the following we will compare the two cases following Fig. 7.11. Born scattering is described by (7.42) where the atomic scattering factors f a express the conversion efficiency of a plane into a spherical wave. In HF scattering, the real atoms are replaced by “points” that convert a plane wave into a spherical wave with unit efficiency, corresponding to a virtual scattering length f v . The size 8 This is formally seen by extending the integration in (7.39) to R → ∞ so that only the sheet scattering remains according to

∞ ik z 02 +r 2 Na f a e Na f a O 2π r dr = −iE 0 λ eikz 0 . E scat = −E 0 π R2 π  R2 2 2 z +r  0

.

0

sheet scattering

(7.41)

338

7 Semi-classical Response of Solids to Electromagnetic Fields plane wave

E0

optical axis

spherical wavelets area A point scatterers

O r

E0 source strength

k

E0

r f

S = E0 f

Fig. 7.11 Illustration of the Huygens–Fresnel principle of conversion of a plane wave of amplitude E 0 and wavelength λ into spherical wavelets (restricted to the forward half of the total 4π solid angle) that originate in a plane of area A shown in gray. The red points represent either atoms in a thin sheet or virtual scattering centers of the same density in an empty plane of size A. We are 0 , interested in the field transmitted on the optical axis which is again a plane wave of amplitude E scat as indicated. In the inset we picture the conversion of a plane wave of amplitude E 0 into a spherical wave E = E 0 f /r , arising from a source of strength S = E 0 f where f is the scattering length. In the text we distinguish f a for an atom and f v for a virtual point scatterer

of the “points” is left unspecified in the HF principle, but from an x-ray point of view one may envision the points as hypothetical spheres of atomic size. We simply replace the real Na atomic scattering centers in the thin sheet by virtual atoms of the same density in a circular hole of the same size A. The HF transmitted field at an on-axis point O is then given by the transmitted Born field (7.42), written in terms of the virtual scattering length f v as O E trans = E 0 eikz0 − i   hole E trans

Na f v λ Na f v λ E 0 eikz0 . E 0 eikz0 = E 0 eikz0 − i   A  A    hole E scat

hole E trans

(7.43)

=1

where the second term now reflects the scattering of the virtual atoms with scattering length f v . On the far right side we have invoked that in HF scattering, there is complete conversion efficiency of a plane into a spherical wave so that the superposition of the spherical waves at an on-axis point must have a value equal to the incident field. Hence we find Huygens–Fresnel scattering length: f v =

A . i λ Na

(7.44)

As expected, the virtual sources of atomic size have a purely imaginary scattering length. In Fig. 7.11 the scattering length is denoted generically as f , and we have f = f a for real atoms and f = f v for virtual atoms. As illustrated in the inset of Fig. 7.11, the scattering length characterizes the efficiency of converting the incident plane wave into scattered spherical waves with a source strength S = E 0 f . Since

7.5 From Dielectric to Atom-Based X-Ray Response

339

E 0 has the dimension [V/m] and f has the dimension [length], the source strength S has the dimension [V].9 We have derived a remarkable result. We can describe the conversion of an incident plane wave into spherical waves by either real atoms in a thin sheet with scattering lengths f a or by virtual atoms of the same density in an open aperture with scattering length f v . The on-axis “transmitted field” through a thin sheet of area A given by (7.42) can be pictured as arising from scattering by the real atoms in the sheet with scattering length f a and Huygens–Fresnel virtual atoms of the same density and scattering lengths f v , according to tot E trans =i

Na λ f v Na λ f a E 0 eikz0 − i E 0 eikz0 . A  A 

  v virtual atoms: E scat

(7.45)

atom real atoms: E scat

The conversion efficiency of the incident plane wave into spherical waves by the real and virtual atoms is determined by the respective scattering lengths f a and f v . The virtual and real scattering contributions have opposite sign, revealing that opposite to the real atoms, the virtual atoms do not induce a 180◦ scattering phase shift. Let us put in some numbers and compare the real atomic (Thomson) scattering length for Fe atoms f a = Zr0  7 × 10−5 nm to that of virtual scattering centers with the same density in a plane. With the atomic density ρa = 84.9 atoms/nm3 for Fe metal, a single layer has a thickness given by the atomic diameter d = 0.23 nm and contains Na /A = 19.3 atoms/nm2 . For a wavelength of 1.75 nm (Fe L3 -edge) we obtain from (7.44) | f v |  3 × 10−2 nm. The Born atomic scattering factor is smaller than the virtual HF scattering factor by | f a / f v |  2 × 10−3 . This number is in good accord with expectations based on the Fe L3 -edge fluorescence yield of 6 × 10−3 [22] which expresses the atomic x-ray emission probability in random directions, classically corresponding to the emission of a spherical wave.

7.5.4 Scattering Phase Shifts The forward scattered wave by the atomic sheet exhibits a different phase than that forward scattered by a single atom discussed in Sect. 6.5. For a single atom, the scat9

As discussed in Sect. 3.2.3, a synchrotron radiation source generates an electromagnetic wave that is strongly forward directed and can be described as a plane wave with a field amplitude E 0 ∼ 100 V/m. If this field scatters off an atom in the middle of the periodic table (Z ∼ 30) with an atomic Thomson scattering length of f = Zr0  10−4 nm, the associated source strength is S  10−11 V.

340

7 Semi-classical Response of Solids to Electromagnetic Fields

tered field for non-resonant Thomson scattering is described by the atomic scattering factor f a = r0 Z + f  − i f   r0 Z , and the forward scattered field is given by the Born approximation (6.49) with r = z 0 as atom E trans =

r0 Z E 0 eikz0 − E 0 eikz0 .

  z0

  plane wave: E in

(7.46)

forw. scatt. wave: E scat

Comparison with the forward scattered Thomson field of an atomic sheet given by (7.42) or sheet = E trans

E eikz0 0   plane wave: E in

Na λ − ir0 Z E 0 eikz0 A

 

(7.47)

sheet Thomson: E scat

reveals a relative phase shift difference of 90◦ (factor i) for the two cases. This is a geometric effect arising from the integration over the atomic plane. For resonant scattering, the atomic scattering factor reduces to the resonant form f a = r0 Z + f  − i f   −i f  (see Fig. 6.5) so that (7.42) becomes Na λ sheet E trans = E 0 eikz0 − f  E 0 eikz0 . A  

(7.48)

sheet Resonant: E scat

Now the forward scattered wave is 180◦ out of phase with the incident field. Our above results apply for scattering by atomic charges and are independent of the incident polarization. We can summarize the scattering phase shifts as follows. For forward scattering by atomic charges, the phase shifts upon scattering are different for non-resonant and resonant scattering and also for scattering from a single atom and an atomic thin sheet. The following statements hold independent of the incident polarization. For non-resonant Thomson scattering, the scattered wave is phase shifted relative to the incident wave by 180◦ for a single atom but by 90◦ for an atomic sheet. For resonant scattering, the scattered wave is phase shifted relative to the incident wave by 90◦ for a single atom but 180◦ for an atomic sheet. For the case of a thin atomic sheet, which is of primary interest in the present chapter, the collective forward scattering phase shift of 180◦ in resonance scattering will result in destructive interference of the weak forward scattered wave with the strong incident wave. This leads to a reduction of the intensity in the forward direction, which we typically associate with a loss due to “absorption”. We will use this important forward scattering loss in Sect. 7.8 to show that one can indeed derive the

7.5 From Dielectric to Atom-Based X-Ray Response

341

exponential absorption law for a thick film by the consecutive addition of thin sheet losses in the forward direction.

7.5.5 Link of X-Ray Scattering Factors and Optical Parameters Having derived the forward scattered fields by an atomic sheet, we are now ready to establish the fundamental relationship between the atomic scattering factors and the optical parameters. In contrast to the optical range, where the small photon energy causes an interaction mostly with the weakly bound valence electrons, the interaction in the x-ray regime is primarily with the atomic core electrons. They typically constitute a larger fraction of the total number of atomic electrons Z and most importantly have larger interaction cross sections as shown in Fig. 1.27. The latter point is beautifully demonstrated by the fact that high-intensity and short x-ray pulses can strip atoms from the inside-out, leaving behind hollow atoms with only the valence electrons remaining [23, 24]. One may therefore formulate the interaction of x-rays with matter in terms of the response of the total of Z electrons per atom and their atomic number density ρa . The atom-based treatment is a very good approximation over most of the x-ray region with the exception of the narrow region around absorption edges, where large resonances are observed that depend on the detailed valence electronic structure. The x-ray scattering factors found in the literature cover only the non-resonant spectral ranges as discussed in Sect. 6.7. They may then be supplemented near absorption edges by inclusion of core-to-valence resonances, either empirically from measured spectra or through quantum mechanical calculations introduced in Chap. 9. We may compare two formulations of the field transmitted through a thin sheet. The first is based on an atomic scattering factor-based approach. It expresses the transmitted Born field as (7.42) or  Born E trans

= E0 e

ikz 0

Na λ fa 1−i A

 .

(7.49)

For forward scattering the atomic scattering factor given by (7.38) simplifies since  ·   = 1 and may be expressed as f a = r0 [ f 1 − i f 2 ]. The atomic area density in the thin sheet can be written as Na /A = ρa , where ρa is the atomic volume density and  the sheet thickness. Then (7.49) becomes Born = E 0 eikz {1 − [i f 1 + f 2 ] r0 ρa λ } . E trans

(7.50)

This expression can now be compared by one given in terms of the refractive index or optical parameters, given by (7.24). By replacing the arbitrary thickness d of the film by   d, the exponential reduces to a linear response expression in δ

342

7 Semi-classical Response of Solids to Electromagnetic Fields

and β, and with k = 2π/λ we obtain   2π  index = E 0 eikz 1 − [iδ + β] . E trans λ

(7.51)

Comparison of the real and imaginary parts in (7.50) and (7.51) gives us the desired fundamental link between optical parameters and the atomic x-ray scattering factors. The optical parameters and scattering factors and lengths are related according to  λ2 ρa

λ2 ρa (7.52) r0 f 1 (ω) = r0 Z + f  (ω) , δ(ω) = 2π 2π β(ω) =

λ2 ρa λ2 ρa  r0 f 2 (ω) = f (ω), 2π 2π

(7.53)

where δ and β ≥ 0 are real and dimensionless numbers, the scattering factors f 1 and f 2 ≥ 0 are in units of number of electrons, and the scattering lengths f  and f  have the dimension [length]. To illustrate the relative size of the scattering factors, scattering lengths, and optical parameters, we show in Fig. 7.12 the values for Fe metal in the L-edge region.

-4

1.0 10-4 5.0 10-5 0 -5

-5.0 10

-4

-1.0 10

60 f’’, f2 ,

40

0.004

20

0.002 0

0 -20

0.006

and

“ (nm)

1.5 10 -4

f ‘+ r0Z and f

0.008

Fe metal

f1 and f2 (Number of electrons)

2.0 10

f ’+ r0Z, f1 ,

-0.002 -0.004

-40

-1.5 10 -4

-0.006 -60 680 690 700 710 720 730 740 Photon energy (eV)

Fig. 7.12 Energy dependent scattering lengths f  + r0 Z and f  (blue), scattering factors f 1 and f 2 (black), and optical parameters δ and β (red) near the L3,2 -edges of Fe metal, related through (7.52) and (7.53). Data are those of Fig. 6.10 from [25]

7.6 The Optical Theorem

343

For a sample composed of different atoms, we can write the following relationship between the complex refractive index n(ω) ˜ in (7.20) and the scattering factors n(ω) ˜ = 1 − δ(ω) + i β(ω) = 1 −

 r 0 λ2   ρ j Z + F j (ω) − iF j (ω) , 2π j

(7.54)

where r0 is the classical electron radius, λ is the wavelength, and ρ j is the number density of atomic species j (atoms/length3 ), so that the right hand side of (7.54) is dimensionless, as required.

7.6 The Optical Theorem The optical theorem expresses energy conservation between the incident beam, said to propagate into the “forward direction”, and that lost in and emerging from the sample in all directions. It was first stated independently in 1871 by Wolfgang Sellmeier and William Strutt (later Lord Rayleigh). The interesting history of the optical theorem has been reviewed by Newton [26]. As the name implies, the optical theorem was originally established in the visible region. Over the years the improved understanding and development of quantum theory have led to its clarification and repeated “reinvention”. Formally, the theorem may be expressed through the imaginary part f  (ω) of the atomic scattering length given by (6.30) as f (Q, ω) = f 0 (Q) + f  (ω) − i f  (ω). In its most general form, the optical theorem reads [26] Optical theorem : Im f (Q = 0) = f  =

k σtot . 4π

(7.55)

where k = 2π/λ. The left side of (7.55) describes the radiation loss (imaginary part) experienced in the forward direction, k = k, corresponding to no momentum transfer Q = 0 (see Fig. 6.3). The total cross section σtot on the right side describes all interactions with the sample that lead to a loss of energy from that re-emerging in the forward direction. Classically, the imaginary part of a response function describes how a physical system dissipates energy when it is out of phase with the driving force. For example, energy may be lost from the photon system to the electronic system (or lattice) so that f  is associated with “absorption” of radiation. Then (7.55) with σtot = σabs is just our previous semi-classical x-ray expression (6.46) 2λ f  = σabs . In the optical region, the response of matter strongly varies with its valence electronic structure, and “absorption” may be defined in different ways [27]. For gases and insulators the energy of visible light is typically less than the ionization potential or band gap. Dissipative processes where energy is irreversibly transferred to the electronic system are then absent. In these cases “absorption” may be defined as loss

344

7 Semi-classical Response of Solids to Electromagnetic Fields

of photons in the forward direction due to out-of-beam scattering into 4π. Hence “absorption” and scattering cross sections are the same, as for the Breit–Wigner cross section (6.56). The “absorption” loss in the forward direction is then compensated by the gain of the energy scattered out of the beam direction, and we have σtot = σabs = σscat in (7.55). The situation is different when electrons in the sample can be excited by visible light as in metals. Then true dissipative energy losses exist, and absorption needs to be distinguished from scattering. In the optical range some of the incident energy is irreversibly lost by transfer to the electronic or phonon system. In the x-ray regime, the incident radiation always has enough energy to create core holes that may be filled through Auger decay. Hence irreversible energy transfer from the photon to the electron system always exists. Then σtot in (7.55) takes the interesting general form σtot = σabs + σscat . In the following section we will discuss this interesting case and show that, in practice, the optical theorem in the x-ray range is described by Im f (Q = 0) =

1 σabs k σtot = . (σabs + σscat )  4π 2λ 2λ

(7.56)

The right side then expresses the fact that in the x-ray region absorption completely dominates so that the scattering cross section σscat in (7.56) can be neglected and (6.46) is obtained.

7.7 Coherent Versus Incoherent X-Ray Scattering from Solids In our previous discussion of x-ray scattering we considered the special case of forward scattering along the optical axis. In the classical description of x-ray scattering, the fields scattered into the forward direction are in phase corresponding to their coherent superposition. More generally, the field scattered into an arbitrary direction by a solid sample depends on the relative phases between the fields scattered by the individual atoms, and one can distinguish two general cases. Incoherently scattered fields have random relative phases and their intensity (field squared) scales linearly with the number of scattering centers. Coherently scattered fields have the same phase and their intensity scales quadratically with the number of scattering centers.

7.7 Coherent Versus Incoherent X-Ray Scattering from Solids

345

7.7.1 Incoherent Versus Coherent Field Superposition Let us consider how the scattered fields from Na atoms in the incident beam add up to the scattered intensity. In our discussion we follow Fig. 7.13a. We are interested in the difference in the intensity scattered into the forward direction k relative to the intensity scattered into a solid angle d around a direction k = k. We may refer to the two cases as “in-beam” and “out-of-the beam” scattering. We consider the scattered intensity from a thin sheet sample containing Na atoms in the illuminated area A, measured by a detector at a distance r that is very large compared to the illuminated sample area. To facilitate our discussion, we start with the two atom case illustrated in Fig. 7.13b. The two atoms are separated by a distance  whose spherical wave fields add at the point of a distant detector D. The path length differences  from the atoms to the detector lead to a phase shift factor exp(ik ), so that the total scattered field at the detector in the Born approximation at the detector position r is the sum of the phase-shifted fields from the atoms.

(a) Forward and out-of-beam scattering 2 2 2 Escat ~ E0 r

x

detector area

dB = d r 2 k’

r

d

sample wavelength 2

E0

A

k

2 Escat

y

z

z

Na atoms

(b) Out-of-beam scattering = incoherent scattering D k’

r

k Fig. 7.13 a Geometry for “in-beam” forward scattering into direction k from an illuminated area A of an atomic sheet versus “out-of-beam” scattering into a solid angle d around direction k . b Simplified case of scattering of an incident plane wave by two atoms separated by a distance  into the direction k of a distant point detector located at a distance r  . At the detector position, the spherical wave fields scattered by the atoms have a phase difference  due to the different path lengths

346

7 Semi-classical Response of Solids to Electromagnetic Fields

For soft x-rays, we can assume that the atomic scattering length f a is independent of the scattering direction. This corresponds to two assumptions. First, that the atomic form factor (6.25) is not Q-dependent, i.e. that the wavelength is much longer that the atomic dimension. Second, that the polarization of the incident light is ignored, i.e. that the fields are treated as scalars and the angular integration simply gives a factor 4π.10 We can readily extend the two atom case to Na atoms by using the label  for the phase shifts of the Na interfering fields and obtain a  E0 eik . f a eikr E scat (r) = − r =1

N

(7.57)

Here we have used the fact that for elastic scattering k  = k and r  l . The latter condition allowed us to approximate r + l  r in the denominator, but we needed to keep the full expression for the phases since they enter exponentially. In linear response, the scattered intensity is given by N 2 a   2 E   |E scat (r)|2 = 20 | f a |2  eik  .   r =1

(7.58)

The phase term differs for coherent and incoherent superposition of the fields. For coherent superposition all the phases are the same and the absolute value of the sum yields Na2 . If the phases are random, only the terms eikn e−ikn = 1 remain while the interference terms eikn e−ikm cancel because of their random nature. We therefore obtain N 2  a   Na2 coherent  ik  e  =  Na incoherent .  

(7.59)

=1

Indicating the incoherently scattered intensity by the superscript “inc”, we obtain inc 2 |E scat | = Na

|E 0 |2 |E 0 |2 dσscat 2 | f | = N . a a r2 r 2 d

(7.60)

In the last step we have used (6.19) to obtain the intensity scattered into a solid angle d around k . For the calculation of the incoherently scattered intensity we can thus ignore the orientation of the film and just calculate the intensity scattered by a single atom and then multiply by the number of atoms Na in the incident beam.

Inclusion of polarization effects would instead give a factor  = 8π/3 since the scattered intensity has a node along the polarization vector.

10

7.7 Coherent Versus Incoherent X-Ray Scattering from Solids

347

By considering the definitions of the different beam parameters, we can write (7.60) in terms of the following general relationship between the intensities and number of photons per solid angle d inc 2 | Iscat n scat A |E scat = = , 2 |E 0 | I0 n 0 d r 2

(7.61)

where n 0 is the number of incident and n scat the number of scattered photons. We can then state as follows. For an incident beam containing n 0 photons within the beam cross-sectional area A that illuminates Na atoms in a sample, the number of photons that are incoherently scattered into d is given by Na dσscat n scat = n0 . d A d

(7.62)

7.7.2 X-Ray Forward Scattering and Absorption The optical theorem links what we refer to as x-ray absorption and “forward scattering”. Forward scattering means that the fields are considered only within the coherence cone of the radiation emerging from the sample (see Fig. 4.7). One may then distinguish the radiation that remains within the incident beam cross-sectional area from that scattered out of the beam. Below we discuss the “in-beam” forward scattered field and intensity and in the next section consider the “out-of-beam” scattered fields. By use of (7.52) and (7.53) we may introduce the convenient short notation 

C = r0 Z + f  −i f  ρa λd = [δ − iβ] kd ,

(7.63)

where C is complex and scales linearly with the sample thickness d. At the exit plane of the sample the field and intensity transmitted into the forward direction are then given by the exponential attenuation law (7.24) which holds to all orders in C or thickness d. At a distance z behind the sample the forward scattered fields are in phase and confined to the coherence cone and we have E trans = E 0 eikz e−i C , Itrans = I0 e−2λ f



ρa d

= I0 e−2βkd .

(7.64)

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7 Semi-classical Response of Solids to Electromagnetic Fields

The exponential in the transmitted field can be expanded to different orders in f  or β. The corresponding intensity will be correct only to the same order as the field. In first order, we obtain in the scattering length notation11  First order : E trans = E 0 e

ikz





[1 − i C] , Itrans = I0 1− 2λ f ρa  .  

(7.65)

σabs

This expression shows that within first order or linear response, the transmitted intensity contains a loss term (minus sign) that can be attributed to absorption, as indicated by the absorption cross section σabs previously given by (6.72). In a scattering picture, the loss term in (7.65) may be interpreted as destructive interference of the incident field with the spontaneously forward scattered field created by radiative decay of a core hole. The process scales linearly with the number of atoms, and for a thin film the scattered field is 180◦ phase shifted as discussed in Sect. 7.5.4. We are now interested in calculating the transmitted intensity to second order in the optical parameters. We then need to start with the transmitted field expressed to second order   (iC)2 ikz −i C ikz  E0 e . (7.66) 1 − iC + Second order : E trans = E 0 e e 2 The transmitted intensity in the scattering length notation is obtained as12 −2λ f  ρa 

Second order: Itrans = I0 e

+



 I0 1 − 2λ f  ρa    1st ord. abs. loss  2λ2 ( f  )2 ρa2 2 .

 

(7.67)

coh. forw. scat. gain = 2nd ord. abs.

The intensity now contains an additional second order contribution that scales with the number of atoms squared Na2 since ρa  = Na /A. It is simply the second order term in the conventional exponential absorption law, and its positive sign corresponds to a gain that is absent in first order. We can summarize as follows.

11

We have ignored diffraction effects from the edge of the illuminated area A, but they can be neglected if we use a large beam diameter. 12 Note that the expression for the second order transmitted intensity (7.67) cannot be obtained if we used only the first order field expression (7.65). If we did so and naively kept the second order

 term, the intensity would have the incorrect second order term λ2 ρa2 2 (r0 Z + f  )2 + ( f  )2 .

7.7 Coherent Versus Incoherent X-Ray Scattering from Solids

349

In first order or linear response, the transmission loss of the intensity scales linearly with the number of atoms Na in the beam. It may be interpreted either as absorption loss, or equivalently, as a loss caused by destructive interference of the incident field with the 180◦ phase shifted field that is spontaneously forward scattered by the atomic sheet. In second order, there is a gain in transmission that scales with Na2 . It may be interpreted either as the second order correction term within the exponential absorption law, or equivalently, as a gain caused by constructive interference of the incident field with an in-phase forward scattered field.

7.7.3 The Total Transmitted Intensity Our above discussion reveals the fascinating correspondence between the first and second order terms in the exponential absorption law and resonant scattering processes in the forward direction. For a more complete description of the radiation emerging from the sample, we need to also consider the fraction scattered out of the forward direction of the beam. For an atom this is described by the atomic scattering cross section σscat given by (6.71) or σscat =

 8π

(r0 Z + f  )2 + ( f  )2 . 3

(7.68)

For a sample with randomly distributed atoms, the fields scattered in different directions will superimpose with random phases and the scattered intensity will scale linearly with the number of atoms Na . In contrast to the “in beam” coherent forward scattered intensity, the out-of-beam scattered intensity is incoherent and given by inc = I0 Iscat

 Na 8π

(r0 Z + f  )2 +( f  )2 . A 3

 

(7.69)

σscat

The factor 8π/3 differs from the factor 4π for isotropic emission and accounts for the fact that the field is not a scalar but a transverse vector field with no elastic scattering into the direction of E. In principle, part of the incoherently scattered intensity is also emitted as a gain into the forward direction of the beam, but we ignore it here. The total intensity emerging from the sample may then be written up to second order in the scattering factors as follows.

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7 Semi-classical Response of Solids to Electromagnetic Fields

The total intensity transmitted by a sample of atomic areal density Na /A = ρa , where  is much smaller than an x-ray absorption length, is given up to second order in the atomic scattering lengths by   8π

(r0 Z + f  )2 +( f  )2 ρa  Itrans = I0 1 − 2λ f  ρa  −   3

  lin. abs. loss out-of-beam loss Thomson+resonant scat.  + 2λ2 ( f  )2 ρa2 2 .

  coh. forw. scat. gain =2nd order abs. term

(7.70) The out-of-beam scattering term has a minus sign, meaning that it corresponds to a “loss” from the in-beam transmitted intensity. We note that in the optical region, where electronic transitions may be absent in atoms or band gap materials, the out-of-beam scattered intensity is sometimes treated like an “absorption” loss [27]. Our treatment of absorption and scattering to the same second order is of fundamental importance since it reveals the different dependence on the number of atoms in the beam in the second order (∝ Na2 ) and first order (∝ Na ) terms. Equation (7.70) will serve as the starting point for the formulation of diffraction imaging in the next chapter and the introduction of stimulated forward scattering in Chaps. 13 and 15.

7.7.4 Relative Size of the Absorbed and Scattered Intensities The cross sections σabs and σscat are the conventional absorption and scattering cross sections for a single atom Na /A = 1, previously given by (6.72) and (6.71) and shown for Fe atoms in Fig. 6.9. We can now update this figure by inclusions of the pronounced resonant contributions at the L-edges for Fe metal from Fig. 7.12, and the result is shown in Fig. 7.14. The figure clearly reveals that absorption dominates at 100 eV by about five orders of magnitude over x-ray scattering, while the difference decreases to less than two orders of magnitude at 10 keV. The response reflects the optical theorem in the form (7.56). Near absorption thresholds there is strong resonance behavior in both absorption and scattering. The details of non-resonant versus resonant response, indicated in red and blue, are best treated quantum mechanically as done in Sect. 9.5. There we also show in Fig. 9.5 an annotated version of Fig. 7.14 that distinguishes the different contributions.

Absorption and scattering cross sections (barn/atom)

7.8 The Response of a Thick Sample: Dynamical Scattering Theory 107

351

Fe Absorption

106

105

104

103

102

resonant processes

L 2,3

Scattering

K

non-resonant Thomson scattering

101

1

100

1000

10,000

Photon energy (eV) Fig. 7.14 Angle integrated absorption and scattering cross sections per atom for Fe metal including the detailed resonant contributions from Fig. 7.12 which are absent in Fig. 6.9. The absorption cross inc in (7.69). Resonant processes section σabs is defined in (7.65) and the scattering cross section σscat near the L2,3 edge are highlighted by a red oval, and we have also indicated in blue the nearly energy independent behavior of Thomson scattering

In the following section we will take the next building block step and add the atomic sheets to a film of arbitrary thickness.

7.8 The Response of a Thick Sample: Dynamical Scattering Theory The transmitted intensity given by (7.70) is valid only for a thin film sample of at most one absorption length thickness. The loss due to true absorption is described through the second and last terms which represent the second order expansion of the exponential absorption law given by (7.32). We will now show that the full exponential absorption law for a sample of arbitrary thickness may be derived by a series of contributions from atomic sheets that are stacked together. The theory is called the Darwin-Prins theory after Darwin [28, 29] who in 1914 developed self-consistent equations that describe wave propagation between periodic layers and Prins [30] who in 1930 extended the theory by including absorption through a complex refractive index.

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7 Semi-classical Response of Solids to Electromagnetic Fields

In scattering theory one distinguishes so-called kinematical from dynamical treatments. The kinematical approximation assumes that the scattering is so weak that the probability of multiple scattering events in a sample can be neglected. With increasing probability of multiple scattering events the dynamical scattering limit is approached. If we were concerned with conventional x-ray scattering by a thick sample, kinematic scattering theory would typically be sufficient because of the weak atomic scattering cross sections, corresponding to the third term in (7.70) and illustrated in Fig. 7.14. In contrast, the transmission through a thin atomic sheet is dominated by the second term in (7.70). It reflects the first order absorption response or in a scattering picture a coherent forward scattering event where the incident field interferes with the spontaneously 180◦ phase shifted forward scattered field. This type of absorption or coherent scattering event is not weak. When considered sequentially through a stack of sheets, it thus requires a multiple scattering treatment that constitutes a particularly simple and beautiful example of dynamical scattering. The essence of the Darwin-Prins treatment is that by keeping the sheet thickness small one only needs to consider the first order or linear response of the field given by (7.65). Multiple scattering in a sample of arbitrary thickness d = N  is then built up through sheet-by-sheet single coherent forward scattering events, up to the total number of sheets N . Since we have not previously discussed the x-ray reflectivity of solids, we start with a brief review of Snell’s reflectivity law from a classical point of view, based on how the propagation of a plane wave is changed in a dielectric continuous medium. The other sections are devoted to a discussion of x-ray reflection and transmission from an atomic scattering/absorption point of view for a sample of finite thickness.

7.8.1 Snell’s Law and Total X-Ray Reflection Snell’s law is named after the Dutch astronomer and mathematician Willebrord Snellius (1580–1626), although it is now known that the law was first discovered by the Persian mathematician and physicist Ibn Sahl and documented in an optical treatise in 984 [31]. It is typically derived from purely classical considerations, namely how a plane EM wave responds to a change of the complex refractive index n. ˜ From a modern point of view, this treatment is based on two key assumptions: (i) that the x-rays can be described as a classical plane EM wave and (ii) that the sample can be treated as a continuous (homogeneous) medium, with all sample properties lumped into a single parameter, n. ˜ In the x-ray range this problem is greatly simplified by the fact that according to (7.20), the real part of the refractive index n(ω) = 1 − δ(ω)  1, where δ is strongly energy dependent as plotted for Fe in Fig. 7.5 and for polystyrene in Fig. 7.7. Snell’s law is illustrated in Fig. 7.15 for non-resonant x-rays traversing from vacuum into a material. We follow the convention usually used for x-rays [9, 13, 14] with the beam incident at an angle θ measured from the surface plane. With the geometric definitions in

7.8 The Response of a Thick Sample: Dynamical Scattering Theory

353

n=1

’

~ =1- +i n’ n’

z=0

z Fig. 7.15 Illustration of Snell’s law for x-rays incident at an angle θ from vacuum with n = 1 onto a dielectric with complex refractive index n˜  = 1 − δ + iβ and real index n  = 1 − δ. It is customary in the x-ray region to define the incidence angle θ as measured from the surface plane of the sample. As shown in Fig. 7.5, we √ have δ > 0 except near resonances. When the incidence angle θ is less than a critical angle θc = 2δ, the x-rays are totally reflected

Fig. 7.15, Snell’s law reads n cos θ = = 1 − δ(ω) . cos θ  n

(7.71)

For non-resonant x-rays we have δ > 0 and the incident beam is refracted away from the surface normal as shown. Therefore at small grazing incidence angles that are below a critical angle, θ ≤ θc , x-rays are totally reflected. The critical angle θc in units of [rad] is determined by the real part of the refractive index according to [9, 14] θc =

√ 2δ .

(7.72)

7.8.2 Darwin-Prins Dynamical Scattering Theory In our discussion of the transmitted and reflected intensity from a solid, we want to go beyond the simple continuous medium description of the sample and consider an atom-based response. We will, however, maintain that the sample consists of homogeneously distributed atoms. We will classically describe the x-rays as plane waves, keeping in mind that this is equivalent to assuming that the x-rays are completely coherent with infinite lateral and longitudinal coherence lengths. In practice this is only fulfilled if the incident field is laterally and longitudinally coherent over the illuminated sample volume. We consider a soft x-ray beam of wavelength λ  2 nm, traversing a ultrathin sheet of thickness   0.2 nm, corresponding to the thickness of a single atomic layer, as illustrated in Fig. 7.15b. The soft x-ray condition λ   simplifies our treatment since Bragg diffraction between atomic planes does not exist and Compton scattering can be neglected. We describe the atomic response by scattering factors f (Q) = f a that are angle independent. This is a good approximation below about 1000 eV [32]. We restrict our discussion to the case θ  10◦ , so that we do not have to worry about total x-ray reflection and can conveniently use the same propagation direction and angle

354

7 Semi-classical Response of Solids to Electromagnetic Fields

T0 = E0

m=1 m=3

S0

N sheets

Tm

Sm

Sm+1

Tm+1

m

m +1

m=N

TN Fig. 7.16 Geometry and definition of the incident field E 0 and the reflected (denoted S) and transmitted (denoted T ) fields for a sample composed of N identical thin sheets of thickness   λ. On the right we show the definition of the fields inside the solid for two arbitrary layers m and m + 1 with a center-to-center separation 

θ outside and inside the sheet. This also applies when we stack sheets to form a thin film of thickness d = N , where N is the number of sheets, as shown in Fig. 7.16. We choose the surface normal as our z direction with z = 0 chosen right before the top layer. We start by considering the field scattered by a single sheet in the Born approximation. For our definition of θ shown in Fig. 7.16, the geometrical phase change between two successive sheets is given by φ=

2π  sin θ = k sin θ . λ

(7.73)

The transmitted field at the position  behind the layer is given by the incident field plus the forward scattered field, and according to (7.50) we have 

E 1trans (z = )

Na λ f a = 1−i A

 E 0 (z = 0) eiφ

= (1 − i ρa λ f a ) E 0 (z = 0) eiφ .

(7.74)

The exponential factor accounts for the geometrical path length difference or phase shift upon propagation by the distance . Similarly, the reflected field is given by the back-scattered field by the layer, and since the layer scatters equally in the forward and backward direction, we have at the position z ≤ 0 before the layer E 0refl (z = 0) = −i ρa λ f a  E 0 (z = 0) .

(7.75)

We have distinguished transmitted and reflected fields, and for clarity have used subscripts 0 and 1 to denote at which position the fields are considered. Keeping in mind the correlation between subscripts and positions 1 ⇔ z = 0 and 2 ⇔ z = , we can now generalize our above discussion to more than a single sheet.

7.8 The Response of a Thick Sample: Dynamical Scattering Theory

7.8.2.1

355

The Darwin-Prins Difference Equations

We assume that the incident field is a plane wave. This is equivalent to assuming that the field has an infinite lateral and longitudinal coherence length. In order to keep things simple, we shall for brevity use subscripts only and omit the position dependence. Also, we shall use the convenient short form definition (7.63) or C = ρa λ f a = ρa λ r0 ( f 1 − i f 2 ) = (δ − iβ) k .

(7.76)

In our new notation (7.74) and (7.75) become 1 sheet : E 1trans = (1 − i C) E 0 eiφ , and E 0refl = −i C E 0 .

(7.77)

We now add a second sheet. The field transmitted through the first sheet is then reflected by the second sheet. If we denote the reflected intensity at the position in-between the sheets as E 1refl , the total reflected field by two layers is obtained by considering the transmission of E 1refl back through the first layer and adding it to the field reflected by the first layer, i.e. 2 sheets : E 0refl = −i C E 0 + (1 − i C) eiφ E 1refl .

(7.78)

Instead of being transmitted back through the first layer, the field E 1refl can also be reflected by it. It then contributes to the transmitted field E 1trans at the position between the two sheets, and we have 2 sheets : E 1trans = (1 − i C) eiφ E 0 − i C e2iφ E 1refl .

(7.79)

The last exponential factor accounts for the geometrical path length difference between the two components to E 1trans . Realizing that in our labeling the incident field E 0 can also be denoted E 0trans , we can generalize (7.78) and (7.79) to an arbitrary pair of layers m and m + 1 in a solid. We now change our notation to that often used in the literature [13, 14] Tm = E mtrans ; Sm = E mref ;

trans Tm+1 = E m+1 ref Sm+1 = E m+1 ,

(7.80)

which is illustrated in Fig. 7.16. We obtain the following Darwin-Prins difference equations.

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7 Semi-classical Response of Solids to Electromagnetic Fields

The transmitted and reflected fields associated with two arbitrary layers m and m + 1 as illustrated on the right side of Fig. 7.15b are linked by the Darwin-Prins difference equations according to Sm = −i C Tm + (1 − i C) eiφ Sm+1 Tm+1 = (1 − i C) eiφ Tm − i C e2iφ Sm+1 .

(7.81) (7.82)

These self-consistent equations are valid for soft as well as hard x-rays. For hard x-rays, their solution for a semi-infinite perfect single crystal is the basis for the dynamical theory of x-ray diffraction. The theory determines the important Darwin width of the monochromatic intensity obtained by Bragg diffraction from a perfect crystal. For soft x-rays, where Bragg diffraction does not occur, the above equations can be directly used to calculate the reflectivity of a solid of arbitrary thickness, and we will do so below. We can significantly simplify these equations under the following assumptions: • Normal incidence θ = 90◦ , so that φ = k. • The atoms are uniformly distributed within the sample, and the incident wavelength is larger than the distances between atoms. This eliminates Bragg diffraction. • The scattering process maintains a well-defined phase relationship between the incident and scattered field. This is fulfilled only for elastic scattering where scattering by each sheet leads to a phase shift of 180◦ (minus sign) relative to the incident field as derived in Sect. 7.5.4.

7.8.3 The Transmitted Field and Intensity The Darwin-Prins difference equations readily yield the transmitted field. To see how this works, we calculate the transmitted field for an increasing number of stacked sheets. In our notation, the incident field before sheet 1 is T0 = E 0 .

(7.83)

and the field after sheet 1 in the absence of a second sheet is T1 = (1 − i C) eik E 0 .

(7.84)

If we add a second sheet, the transmitted field after sheet 1 is changed only slightly by the reflected intensity from sheet 2 that is again reflected by sheet 1 to propagate into the direction of the transmitted field. We have according to (7.82) T1 = (1 − i C) eik E 0 − i C e2ik S1 .

(7.85)

7.8 The Response of a Thick Sample: Dynamical Scattering Theory

357

But from (7.82) we can see that the reflected contribution S1 is small. It is given by S1 =

S0 + i C T0  (S0 + i C T0 ) e−ik  0 , (1 − i C) eik

(7.86)

where we have used S0 = −i C E 0 . Hence we can neglect the reflected contributions from now on. With T1  (1 − i C) eik E 0 ,

(7.87)

we get the field after 2 sheets as T2  (1 − i C) eik T1  (1 − i C)2 e2ik E 0 .

(7.88)

As we progress through additional layers, we get another multiplicative transmission factor (1 − i C) exp(ik) for each layer, and after N sheets we have TN  (1 − i C) N e N ik E 0 .

(7.89)

We see that each sheet contributes a multiplicative factor (1 − i C) eik , where C is given by (7.76). Since the absolute value of C is of order 10−2 (see Fig. 7.12 for Fe), we can make the substitution (1 − i C) N = (e−iC ) N in (7.89). In fact, a plot of the absolute values of the two expressions as a function of N shows that they are indistinguishable from each other. Hence for the total film thickness d = N  we obtain with (7.76) the transmitted field in terms of the optical parameters as TN = E trans  E 0 eikd e−i(δ−iβ)kd = E 0 eikd e−iλr0 ( f1 −i f2 )ρa d ,

(7.90)

which is identical to our previous result (7.23). For a film of finite thickness d = N , the linear transmission response due to forward scattering from each sheet of thickness  evolves into an exponential response due to propagation phase shifts between the N sheets. The transmitted intensity is obtained as Itrans  I0 e−2βkd = I0 e−2λr0 f2 ρa d ,

(7.91)

which is the same as our previous derived absorption law (7.32). We see that the link of absorption and coherent forward scattering discussed for a thin sheet in Sect. 7.7.2 can be extended to a thick sample, if we consider dynamical coherent atomic forward scattering, with the scattering of each layer treated by the first order Born approximation.

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7 Semi-classical Response of Solids to Electromagnetic Fields

7.8.4 The Reflected Field in the Soft X-Ray Region Similarly, the total reflected field S0 before the sample is calculated from the DarwinPrins equations by adding the (small) contributions from the N elementary sheets, with the total sample thickness given by d = N . We find for the dependence of the reflected field as a function of the number of sheets N = 1 : S0 = −i C E 0 N = 2 : S0 = −i C E 0 + (1 − i C) eik S1

 N = 3 : S0 = −i C E 0 1 + (1 − i C)2 e2ik + (1 − i C)2 e2ik S2 . (7.92) For an arbitrary number of N sheets we have  S0 = − i C E 0

N 

 (1− i C) e

2 2 ik

+(1− i C) N −1 e(N −1)ik S N −1 .

(7.93)

=0

The reflected field S N −1 arises from the last sheet N . According to (7.82), it is given by S N −1 = −i C TN −1 + (1 − i C) eik S N = −i C TN −1 ,

(7.94)

where we have made use of the fact that there is no reflected field S N from the vacuum behind sheet N . By use of (7.89) we have TN −1  (1 − i C) N −1 e(N −1)ik E 0 .

(7.95)

This allows us to write the reflected field for N sheets as  N   2 2 ik E refl = S0 = − i C E 0 − i C E 0 (1− i C)2(N −1) e2(N −1)ik (1− i C) e =0

 (β + iδ)k   E 0 2 ik 1 − e2ik N  [1 − i(δ − iβ)k]2N , e −1

(7.96)

where in the last step we have expressed C by (7.76), evaluated the sum as a power series in the optical parameters, and kept only the first order terms. For   λ/2π or k  1 we obtain our final result.13

13

This agrees with the result in [13] for an N layer film, given in terms of the reflectivity of a semi-infinite film, given by (7.98).

7.9 Polarization Dependent Absorption: Dichroism

359

The reflected field of a film of total thickness d = N  at normal incidence is given by E refl  E 0

 δ− iβ  1 − e2ikd(1−δ+iβ) . 2

(7.97)

For a semi-infinite solid (d → ∞) the reflected intensity ∞ |E refl |2 oscillates because of the exponential term but assumes the limiting value Irefl  I0

δ2 − β 2 . 4

(7.98)

Because of the small value of the optical parameters, which according to Fig. 7.12 are δ, β < 10−2 even in the resonance regions, the reflected intensity is very small. It is therefore neglected in the exponential absorption law.

7.9 Polarization Dependent Absorption: Dichroism We start this section with an historical overview of the discovery of dichroic or polarization dependent effects which occurred by use of visible light. We then discuss a simple classical description of the experimentally observed effects based on an effective medium response through phenomenologically modified optical parameters. This formalism, which may also be extended into the x-ray region, however, fails to link the effects to the atomic, electronic, and spin dependent structure of matter. This requires quantum theory as will be discussed in Chap. 11 and Sect. 13.6.

7.9.1 History of Polarization Dependent Effects Today the term dichroism refers to the change of the polarization of light upon transmission or reflection by matter. The first documented observation of a dichroic effect goes back to 1669 when the Dutch scientist Erasmus Bartholinus (1625–1698) observed the phenomenon of double refraction in calcspar (calcite). In his 1690 book Traité de la lumiére, Huygens reported that the two rays arising from refraction in calcite may be extinguished when passing through a second calcite crystal. The next milestone came another 100 years later, when in 1808 the Frenchman Étienne Louis Malus (1775–1812) used calcite to show that light may become polarized upon reflection by a glass window pane. Malus also first used the term “polarization” but did not attempt to interpret the phenomenon. In 1811 Arago [33] discovered the phenomenon of optical activity, consisting of a rotation of the polarization direction upon transmission through crystalline quartz, and around the same time Biot extended the measurements to vapors and liquids [34]. In 1815 the Scottish physicist David

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7 Semi-classical Response of Solids to Electromagnetic Fields

Brewster (1781–1868) discovered the relationship between the index of refraction and the angle of total reflection resulting in complete linearly polarized light. The phenomenon of light polarization was put on a firm scientific footing by Fresnel in the period 1818–1822. He suggested that light is a transverse wave and introduced and linked the concepts of linearly and circularly polarized light, discussed in Sect. 3.2.6. Fresnel also attributed optical activity to different indices of refraction of the left and right components. Fresnel’s explanation was supported by an important experiment carried out by Louis Pasteur in 1850 [35]. He observed that the organic compound tartrate, when synthesized in a laboratory, was optically inactive, unlike the tartrate extracted from grapes. He then succeeded in separating the left- and right-handed crystals from each other and showed that each was optically active with opposite rotations of the polarization. Despite Fresnel’s general explanation of optical activity and Pasteur’s recognition that chiral molecules give rise to optical activity, it was not until 1895 that Cotton [36] performed what might be called the first circular dichroism experiment. He discovered that natural light becomes partially circularly polarized when traversing a chiral medium. The importance of the concepts of handedness or chirality is reflected by the fact that all 20 biologically active amino acids are left handed, while amino acids made in the laboratory are 50% left and 50% right handed. Owing to the fact that many drugs need to be synthesized in a single chiral state, the topic of optical activity has remained pertinent in the fields of synthetic and structural chemistry, biochemistry, and in pharmaceutical science [37]. Originally, the observation of light polarization effects was restricted to nonmagnetic samples. In 1846 Faraday discovered a direct connection between magnetism and light polarization, the magneto-optical or Faraday effect [38]. It refers to the change of light polarization in transmission through a magnetized material. The same effect in reflection was discovered in 1877 by the Scottish physicist John Kerr (1824–1907) [39] and is called the magneto-optical Kerr effect in his honor. As discussed earlier, these magneto-optical effects do not arise from the interaction of the EM H-field with the sample but indirectly from interaction of the E-field which couples to the charge and then indirectly via the spin-orbit interaction to the spins or magnetization. Today, the magneto-optical Kerr effect (MOKE), which because of its reflection geometry is widely applicable, remains an important tool in magnetism research [40, 41]. In addition, the availability of short and intense optical laser pulses has created the new field of ultrafast magnetization dynamics [42, 43].

7.9.2 Chiral Versus Magnetic Orientation The key difference of polarization effects observed for chiral and magnetic materials results from the basic symmetry concepts of inversion symmetry (parity) and time reversal symmetry. This is beautifully demonstrated by the conceptually simple light transmission experiments through a chiral and magnetic sample shown in Fig. 7.17.

7.9 Polarization Dependent Absorption: Dichroism

361 (b) Magnetic Sample

(a) Chiral Sample

Mirror

Mirror E right handed

right handed

E

Magnetic sample

Chiral sample EM wave

EM wave

Polarizer

E

M

Polarizer

E

M

E

2

Fig. 7.17 Comparison of transmission of a linearly polarized electromagnetic wave through a right-handed chiral in (a) and a magnetic sample in (b). In both cases the linearly polarized wave experiences a rotation of its polarization upon transmission through the sample by rotation angles α and θ, respectively. The beams are then reflected by a mirror that preserves the polarization, and sent back through the respective samples. The right-handed chiral sample always rotates the polarization clockwise when looking into the propagation direction (thumb direction) so that the polarization vector retraces its original path after reflection and arrives back unchanged. The magnetic sample also rotates the polarization clockwise on its first pass through the sample with magnetization direction M pointing into the propagation direction. After reflection the propagation direction is opposite to M and the polarization vector rotates anticlockwise. The light arrives back with the polarization vector rotated by 2θ

In Fig. 7.17a, linearly polarized light is transmitted through a right-handed chiral sample. The handedness is illustrated by the right hand rule, with the thumb pointing into a given direction and the fingers indicating the rotation direction of the helix (see Sect. 3.2.6). The chiral sample causes a rotation of the polarization of the transmitted light by an angle α. The wave is reflected by a mirror which preserves the linear polarization direction. Close inspection reveals that the reflected wave also encounters a right-handed helix, and therefore, the polarization is rotated back into the original direction. This example shows that the chirality of a sample is preserved upon its rotation in space. Hence optical activity may also be observed in liquids composed of randomly rotated molecules of the same chirality. As shown in Fig. 7.17b, a magnetic sample looks different to the EM wave on its first and second pass. The magnetization M is a vector, and it matters whether this vector points into or opposite to the propagation direction of the EM wave. If the polarization vector rotates clockwise by θ on the first path, it will rotate anticlockwise by θ on the second path. The two E-vector rotations by the sample add up, and the final rotation angle will be 2θ. The different behavior of the chiral and magnetic samples in Fig. 7.17 is a direct consequence of the opposite symmetry properties of the two samples. Chirality is

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7 Semi-classical Response of Solids to Electromagnetic Fields

associated with a spatial twist of the charge best described by a helix. A chiral sample lacks a center of inversion. In contrast, a magnetic direction is defined through the direction of current flow through a wire loop. The existence of a magnetic field direction hence corresponds to breaking the time reversal symmetry of current flow. The existence of chirality is therefore linked to the symmetry concept of “space reversal” symmetry or parity and a magnetic direction to the symmetry concept of “time reversal” symmetry.

7.9.3 Fundamental Forms of X-Ray Dichroism On a microscopic level the origin of dichroic behavior of a material originates from the spatial anisotropy of the charge or the spin. In cases where the dichroic effect depends on the charge, only, one speaks of charge or natural dichroism. X-ray “natural” dichroism is caused by the spatial anisotropy of the charge in the absence of spin alignment. – X-ray natural linear dichroism—XNLD—arises from a spatially anisotropic charge distribution. – X-ray natural circular dichroism—XNCD—is associated with a spatial handedness of the charge distribution.

The dichroic effect originating from spin orientation is referred to as magnetic dichroism. In general one distinguishes the existence of a preferential spin direction as in a ferromagnet or ferrimagnet and the existence of an alignment axis, where the same number of spins point in opposite directions as in an antiferromagnet. There are two important types of magnetic dichroism. X-ray “magnetic” dichroism is due to spin alignment and the coupling of the spin to the charge via the spin-orbit interaction. – X-ray magnetic linear dichroism—XMLD—arises from an anisotropic distortion of the charge induced by a preferred spin direction or axis. It exists for all matter with an intrinsic spin orientation or an H-field induced spin orientation. – X-ray magnetic circular dichroism—XMCD—arises from the existence of a preferred spin direction. It exists in ferromagnets and ferrimagnets and paramagnets subjected to a large H-field.

In the following we shall briefly discuss the first x-ray demonstrations of the different effects illustrated in Fig. 7.18, which complements our earlier Fig. 1.19.

7.9 Polarization Dependent Absorption: Dichroism

363

L3

X-ray Magnetic Linear Dichroism

La1.85Sr0.15CuO4 Cu L-edges

Absorption Intensity

(a)

E L2

920

940 Photon energy (eV)

960

6 3 0 -3

6230 6210 5190 Photon energy (eV)

Dichroism (%)

Absorption Intensity

(b)

LaFeO3 Fe L2 -edge

722 724 Photon energy (eV)

X-ray Magnetic Circular Dichroism

X-ray Natural Circular Dichroism LiIO3 I L1-edge

(c)

720

L

Absorption Intensity

Absorption Intensity

X-ray Natural Linear Dichroism

(d)

Fe metal L - edges

720 700 Photon energy (eV)

740

Fig. 7.18 Four important types of dichroism. a X-ray natural linear dichroism spectra of La1.85 Sr 0.15 CuO4 near the Cu L-edge [44]. The resonances are due to transitions to the highest energy unfilled dx 2 −y 2 orbital. b X-ray absorption spectrum (red) of single crystal LiIO3 and the difference spectrum (gray), the x-ray natural circular dichroism spectrum, obtained from absorption spectra with left and right circularly polarized x-rays, incident along a special crystalline axis [45]. c Magnetic linear dichroism spectrum of an epitaxial thin film of antiferromagnetic LaFeO3 with the E-vector aligned parallel and perpendicular to the antiferromagnetic axis [46]. The splitting of the L2 resonance is due to multiplet effects. d X-ray magnetic circular dichroism spectrum around the L3 and L2 edges of Fe metal. The photon angular momentum was aligned parallel or antiparallel to the magnetization direction of the sample [47]

The XNLD effect was first demonstrated in 1981 [48, 49]. Strong XNLD effects are found in the so-called near edge x-ray absorption fine structure (NEXAFS) spectra low-Z molecules, macromolecules, polymers, or nematic liquid crystals [18, 50]. Such systems contain directional covalent bonds leading to polarization dependent xray absorption spectra like that illustrated in Fig. 1.19 for benzene molecules weakly bonded to a Ag(110) surface. In Fig. 7.18a we show the XNLD effect for the Cu L-edge of La 1.85 Sr 0.15 CuO4 [44]. The single crystal sample has a layered structure. The Cu atom shown in black is surrounded by four in-plane O atoms and two outof-plane O atoms. If we define the x, y plane of our coordinate system to lie in the plane of the layers (shown in gray), the in-plane dx 2 −y 2 orbital is unfilled, and when the E lies in the (x − y) plane, a large peak-like transition is observed to this orbital. This resonance is absent when E is oriented perpendicular to the plane, as shown, since there are no empty states in the perpendicular direction. One may envision the polarization vector as a search light for empty valence orbitals, the so-called search light effect [18]. An XNCD effect was first observed in 1998 [45, 51]. It is illustrated in Fig. 7.18b for the iodine L1 -edge x-ray absorption spectrum (red) of single crystal LiIO3 [52].

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7 Semi-classical Response of Solids to Electromagnetic Fields

The XNCD spectrum, defined as the difference of two absorption spectra obtained with left and right circularly polarized x-rays, is shown as a gray curve. It was obtained by aligning the single crystal sample along a special crystalline axis. The effect is seen to be remarkably large, of the order of several percent. In the x-ray region, one distinguishes two types of XNCD effects which arise in higher order beyond the dipole approximation, as will be discussed in Sect. 11.4. The one observed by Goulon and coworkers [45, 51, 52] is only present in oriented crystalline samples and is due to an electric dipole/electric quadrupole interference effect. The other is the x-ray analogue of optical activity and is due to an electric dipole/magnetic dipole interference effect, and it has been studied in [53–57]. The XMLD effect was first predicted and observed in 1986 [58, 59]. A more recent example is shown in Fig. 7.18c at the Fe L2 resonance in LaFeO3 . The spectra show the difference in absorption when the E-vector is aligned parallel and perpendicular to the antiferromagnetic axis in LaFeO3 [46]. XMLD is extensively used today for the study of antiferromagnets, in particular, the determination of the orientation of the antiferromagnetic axis in thin films and near surfaces, and the imaging of antiferromagnetic domains. The principal difference between XNLD and XMLD is that the latter exists only in the presence of magnetic alignment, and hence it vanishes at temperatures above the Neél or Curie temperature, or for paramagnets in the absence of an external magnetic field. The XMCD effect was first reported in 1987 [60]. It is illustrated in Fig. 7.18d for the Fe L-edge in Fe metal. The effect is maximum when the x-ray angular momentum is parallel and antiparallel to the magnetic moment of the sample. The effect is seen to be very large at the resonance positions and is opposite at the L 3 and L 2 edges. This directly reflects the opposite sign of the spin component at the two edges, j = l + s at the L 3 -edge and j = l − s at the L 2 edge. The observation of XMCD requires the presence of a preferred magnetic direction. It is therefore zero for antiferromagnets. There are other more complicated types of “magnetic” dichroism where charge and spin effects are both present [52, 61]. For example, x-ray magnetochiral dichroism arises from axial spin alignment and a chiral charge distribution [62]. X-ray non-reciprocal linear dichroism arises from charge chirality that is induced by an axial spin alignment, and the effect is parity odd and time odd [63]. We shall not discuss these cases here.

7.10 Natural Dichroism and Orientational Order For the phenomenological description of the response of charge distributions in solids to linearly polarized light, i.e. XNLD, one considers the orientation of the E polarization vector relative to different orientations of the sample, assumed to be non-magnetic. In crystalline materials, an anisotropy of the valence charge distribution exists when the lattice has lower than cubic symmetry. A beautiful example of this case is the layered compound La1.85 Sr 0.15 CuO4 whose strongly anisotropic absorption spectrum is shown in Fig. 7.18a. Other examples of strongly anisotropic

7.10 Natural Dichroism and Orientational Order

365

systems are molecules bonded to surfaces, since they typically have a preferred bond orientation [18]. In the following we want to develop a phenomenological description of the general case of orientational order. It means that, on average, a sample contains preferential orientations of interatomic bonds or molecular groups. This may exist even if the sample would appear “amorphous” from an x-ray crystallography point of view. The situation is encountered for many “soft matter” systems including liquids, gels, liquid crystals, polymers, and biological materials. Their exploration was pioneered by Pierre-Gilles de Gennes (1932–2007), who was awarded the 1991 Nobel Prize in physics for his work. Orientational order is conveniently treated by a formalism developed in the optical region [64, 65] that describes charge distributions in terms of their projections onto three axes in a Cartesian coordinate system, so-called orientation factors. Systems without crystallographic order or translational atomic order may nevertheless possess orientational order. It is defined through the existence of a ˆ called “director”, along which there is a preferential spatial symmetry axis u, alignment of internuclear axes or bonds. There may also be additional orientational order in the plane perpendicular to the director, which in its simplest form is called biaxial order. While the spaghetti-like chains in bulk polymers typically arrange themselves in random directions, the randomness disappears through confinement in thin films or the presence of a surface or interface. This leads to preferential orientation of the molecular groups forming the chain backbone and attached functional groups, e.g. phenyl rings. The orientation may also be enhanced in a specific direction by rubbing a polymer. Remarkably, this low-tech technique has been extensively used to create a template for the alignment of liquid crystals, utilized to modulate light transmission in high-tech flat panel displays [50, 66] (also see Fig. 11.6). In the following we shall first discuss how orientational order in rubbed polymers may be detected by XNLD and quantified by a phenomenological framework [50, 67, 68]. Figure 7.19a and b illustrates an XNLD absorption measurement of a rubbed polyimide film, produced for the alignment of liquid crystals in flat panel displays by Japan Synthetic Rubber [68]. The geometry of the experiment is defined in (a) and the NEXAFS spectra for three orientations of the E polarization vector relative to the coordinate system are shown in (b), indicated by corresponding colors. The spectra were recorded by Auger electron yield detection which is preferentially sensitive to the surface of the rubbed polymer film [18, 50]. The observed peaks labeled π1 , π2 , and π3 correspond to specific electronic transitions on the identified C atoms to empty molecular π ∗ orbitals [68] (also see Fig. 1.18). From the angular dependence of the peak intensities one can derive the preferential orientation of the corresponding π orbitals. For example, peak 1 originates from 1s→ π ∗ electronic transitions on the central four C atoms in the phenyl rings and has maximum intensity when the E-vector is parallel to the phenyl π system, i.e. when

366

7 Semi-classical Response of Solids to Electromagnetic Fields

(a)

(b)

JSR-polyimide

O

O

N n

z

E

20o

E y

E x

Absorption intensity

O

N

Rubbing Direction

C C C C

1

1

2

3

289 285 287 Photon Energy (eV)

291

-90 -70 -50 -30 -10 10 30 50 Angle (deg)

70 90

+

y

z

1.0

0.8

1

(d)

peak intensity

1.2 (e)

E x

C N

N C

0

z

N

N

283

(c)

C =O

2

O

E x

y

0.6

Fig. 7.19 Molecular orientation of functional groups with a polyimide film of the structure shown in (a) rubbed in the x-direction. b X-ray absorption spectra for the three the E polarization vector orientations in (a) [68]. The three identified resonances correspond to K-edge transition 1s → π ∗ to empty π ∗ orbitals associated with the identified C atoms in the polymer. c and d define the E-vector orientation angles θ in two orthogonal planes used for the measurement of the π1 peak intensity plotted in (e). We have also identified the three E orientations for the spectra in (b) by bigger symbols with the corresponding colors

E is perpendicular to the plane of the phenyl ring [18]. The figure demonstrates that all π orbitals are preferentially oriented along the surface normal (blue), and there are more π orbitals oriented along y (green) than along the rubbing direction x (red). A more detailed dependence of the π1 resonance intensity as a function of the E orientation is shown in Figs. 7.19c–d as a function of tilt angle θ of E from the surface normal z in the (x − z) (light gray) and (y − z) (light green) planes. The curves in (e) shown in green and black colors are theoretical fits to a model we shall discuss now.

7.10.1 Orientation Factors and Saupe Matrix Orientational order may be treated by concepts originally developed in the late 1950s by Maier and Saupe for the description of the interactions and ordering within liquid

7.10 Natural Dichroism and Orientational Order

367

crystals (LCs) [69]. The formalism is based on the introduction of a tensor order parameter, also called ordering matrix or Saupe matrix, which has been extensively used for the description of the orientational order of nematic LCs [64, 65, 70], polymer surfaces [50, 67, 68], and even amorphous carbon surfaces [71]. We will see in Sect. 11.3.2 that this phenomenological description is closely related to the quantum formulation of XNLD in terms of a quadrupolar distribution. The tensor order parameter description is based on the following assumptions for the “soft matter” system. • The system consists of individual molecular units which are anisotropic (e.g. chains and rings). • The centers of the molecular units exhibit no long range structural order. • On average, the molecular units have a preferred spatial orientation along a symˆ called the director. metry direction u, • The alignment along uˆ and −uˆ is indistinguishable. • There may also be orientational order in the plane perpendicular to the director. In general, the average distribution of the molecular units in the LC is described by a molecular distribution function O(ϑ, ϕ) which depends on the polar and azimuthal angles ϑ and ϕ in a molecular Cartesian coordinate system (x  , y  , z  ) defined below. It is often sufficient to characterize the molecular distribution by the projections of O(ϑ, ϕ) onto the axes (x  , y  , z  ). The normalized projections are called orientation factors [65]. This formalism is suited for the description of polarization dependent optical and XNLD experiments like those shown in Fig. 7.19. XNLD measurements cannot determine the complete distribution function O(ϑ, ϕ) but only its projection. In practice, the molecular frame is typically rotated relative to a Cartesian frame (x, y, z) fixed to the macroscopic shape of the sample. The two reference frames are in general linked by three Euler rotation angles describing a matrix rotation. For our rubbing case, the sample and molecular frames are linked by only a single rotation angle γ , as shown in Fig. 7.20a. The sample frame is chosen with z along the surface normal and x along the rubbing direction. For an unrubbed film the sample frame (x, y, z) and molecular frame (x  , y  , z  ) are the same. The orientation factors in the surface plane are the same, f x = f y , but typically differ from the one along the surface normal, f z as illustrated in Fig. 7.20b. The rubbing process leads to a rotation of the molecular orientation function about the perpendicular y axis, and the sample and molecular frames are now different as shown in Fig. 7.20c. This manifests itself in Fig. 7.19b by an inequivalence of the measured XNLD intensities along x, y and approximately along z, shown as red, green and blue curves. Also, the angular change about the z direction in the (x − y) plane shown as the black curve in Fig. 7.19e is asymmetrical, while it is symmetrical about z in the (y − z) plane as shown by the green curve in Fig. 7.19e. For the rubbed polymer, XNLD measurements determine the orientation factors f x  , f y  , f z  onto the molecular frame axes (x  , y  , z  ) as shown in Fig. 7.20c. They are defined as the projections of the bond distribution function O(ϑ, ϕ) according to 2ππ fx =

sin2 ϑ cos2 ϕ O(ϑ, ϕ) sinϑ dϑ dϕ 0 0

(7.99)

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7 Semi-classical Response of Solids to Electromagnetic Fields

(b) Orientation factors (a) Sample frame x,y,z (c) Orientation factors & molecular frame x’,y’,z’ at natural polymer surface at rubbed polymer surface

fz

z z’

z fz’ fz= fx =fy

y=y’

x’

fx

fz’ = fx’ = fy’

fy

fy’

fx’ x

x

Fig. 7.20 a Sample reference frame (x, y, z), defined by the macroscopic sample shape and rubbing direction, versus molecular frame (x  , y  , z  ) defined by the diagonal Saupe matrix (7.102). b Typical orientation factors f z = f x = f y for unrubbed polymer surfaces for which the sample and molecular frames are the same. c Orientation factors f z  = f x  = f y  in the molecular frame for a typical rubbed polymer surface, where the Saupe matrix is given by (7.105)

2ππ f y =

sin2 ϑ sin2 ϕ O(ϑ, ϕ) sinϑ dϑ dϕ

(7.100)

0 0

2ππ f z =

cos2 ϑ O(ϑ, ϕ) sinϑ dϑ dϕ .

(7.101)

0 0

One can envision these orientation factors as the number of bonds along the molecular frame x  , y  , and z  axes, respectively, and for a normalized distribution  O(ϑ, ϕ) d = 1 we have f x  + f y  + f z  = 1. In the molecular frame one can define a diagonal ordering or Saupe matrix according to ⎛1

(3 f x  − 1) 0    1 ⎝ Q(x , y , z ) = (3 f y  − 1) 0 2 0 0 ⎛ 1 − 2 (S + η) 0 =⎝ 0 − 21 (S − η) 0 0

⎞

0 0

2

1 (3 f z  2

⎞

0 0⎠ . S

⎠ − 1) (7.102)

Here we have introduced the uniaxial order parameter S and the biaxiality η [65] according to 1 S = (3 f z  − 1) (7.103) 2 and

7.10 Natural Dichroism and Orientational Order

η=

369

3 ( f y − f x  ) 2

(7.104)

For η = 0 the distribution is uniaxial and cylindrically symmetric about the z  axis. For finite η the system has different projections along all three Cartesian axes, and we have a biaxial system. For reference, typical nematic liquid crystals are described by a uniaxial distribution η = 0 and order parameters in the range 0.4 < S < 0.7 [70]. In general, the Saupe matrix is not diagonal and one may define the molecular frame (x  , y  , z  ) as that where the Saupe matrix is diagonal. Since the molecular frame is rotated from the sample frame (x, y, z), the Saupe matrix will no longer be diagonal in the latter. For our case it is given by ⎛ ⎜ ⎜ ⎜ ⎝

Q(x, y, z) = − 21 (S +η) cos2 γ + S sin2 γ

0

0

− 21 (S −η)

0

0

− 21 (S +η) sin2 γ + S cos2 γ

1 (S +η) sinγ 2

cosγ + S sinγ cosγ

1 (S +η) sinγ 2

cosγ + S sinγ cosγ

⎞ ⎟ ⎟ ⎟, ⎠

(7.105) where γ is defined in Fig. 7.20a. In general, the Saupe matrix is real, symmetric, and traceless, and the trace is invariant under a unitary transformation which transforms one coordinate system into another. These properties can be verified by inspection of (7.102) and (7.105).

7.10.2 Determination of Orientation Factors The angular plot of the measured intensities in Fig. 7.19e reveals that the scans in the perpendicular planes (black and green) do not have the same value for θ = 0 (z direction). This is due to the fact that data were recorded on a wiggler beam line which produces slightly elliptically polarized radiation. For convenience we used the description of the degree of linear polarization given by (3.35) which in our case was P = 0.82 [68] (see blue curve in Fig. 2.26). If the sample is rotated so that E lies in the (x − z) sample plane which contains the rubbing direction, the XNLD intensity is given by [50, 68] 

I x z (θ ) = I0 A + B  sin2 θ + C  sin2θ A = P ( f x  sin2 γ + f z  cos2 γ ) + (1− P) f y     B  = P ( f x  − f z  ) cos2 γ − sin2 γ tan2γ  C = − B . 2

(7.106)

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7 Semi-classical Response of Solids to Electromagnetic Fields

When E is rotated in the (y − z) plane perpendicular to the rubbing direction, the intensity distribution is

 I yz (θ ) = I0 A⊥ + B ⊥ sin2 θ     A⊥ = P f x  sin2 γ + f z  cos2 γ + (1− P) f x  cos2 γ + f z  sin2 γ   (7.107) B ⊥ = P f y  − f x  sin2 γ − f z  cos2 γ . In practice, one fits the data to derive the parameters A , B  , C  , A⊥ , B ⊥ which also yields γ through the last expression in (7.106) and then calculates the orientation factors according to

fx = f y = f z = Itot =

$ A⊥ + B  1 +

sin2 γ P cos 2γ

% (7.108)

Itot A + B ⊥ Itot

(7.109)

$ A⊥ + B  1 −

cos2 γ P cos 2γ

%

Itot

3  3P − 1  (A + A⊥ ) + (B + B ⊥ ) . 2 2P

(7.110)

(7.111)

The last expression follows from the normalization condition f x  + f y  + f z  = 1. The polarization factor can also be directly obtained as P=

A

B⊥ − B − A⊥ + B ⊥ − B 

(7.112)

The orientation factors for the rubbed PI surface deduced from experiment are summerized in Fig. 7.21a. The molecular π bonds become asymmetrical in the rubbed surface plane and also exhibit an out-of-plane tilt angle γ , as indicated in the figure by the orientation of the yellow π orbital of a phenyl ring. When a liquid crystal is deposited on the surface, the π system of the LC locks to the π system of the polymer surface, which has been termed a “maximum overlap model” of the π charge densities [50]. The preferential bond orientation in the polymer surface acts as a template for LC alignment as shown in Fig. 7.21b, and the LC director exhibits both an in-plane orientation along the rubbing direction and an out-of-plane so-called pretilt angle γ , which plays an important role when the LC orientation is rotated by an applied voltage in a LC display [50, 66, 71] (also see Fig. 1.18).

7.11 Magnetic Dichroism and Faraday Rotation

371

(a) Rubbed polymer template

(b) Liquid crystal alignment

z’ z =

o

N C

5.9

fz’ =0.42

fy’ =0.35

fx’ =0.23

C C C C

y= y’ x’ 5.9

Rubbing direction

O C

o

x

director

x’

Fig. 7.21 a Orientation factors deduced from the experimental data in Fig. 7.19 which reveal the asymmetry of π bonds near the rubbed PI surface, as illustrated schematically by the preferred direction of phenyl π orbitals (yellow). b Corresponding alignment of liquid crystal rod-shaped molecules when placed on the rubbed surface. The orientation of the LC molecules results from locking of the LC π system (yellow) to that of the polymer surface

7.11 Magnetic Dichroism and Faraday Rotation Similar to the phenomenological treatment of charge orientation in XNLD, one may describe spin orientation which leads to the XMCD effect. A preferred spin direction is created by use of an external magnetic field. It is required even for ferromagnets to produce a macroscopic magnetization direction by aligning the randomly spontaneously magnetized (through the exchange interaction) microscopic domains. In contrast to the charge case where the “director” only defines an axis, spins are truly directional vectors, reflecting a broken time reversal symmetry. For the description of XMCD one may also extend a semi-classical formalism which was developed to describe the optical Faraday and Kerr effects [41]. The interaction of x-rays and matter happens on the atomic scale through the coupling of the photon and electronic degrees of freedom. Since the magnetic field of the photon plays no role as discussed in Sect. 7.3.3, the interaction is determined by the coupling of the E-vector to the atomic charge. The coupling to the atomic spin is only indirect through the spin-orbit interaction, which aligns the atomic spin and orbital angular momenta to an atomic magnetic moment. In a simple picture, one would intuitively expect that there is a difference in the coupling or absorption for circularly polarized light if its angular momentum points in the same or opposite direction of the atomic magnetic moments. This is indeed the case and is the origin of the XMCD effect. This effect is closely related to the magneto-optical Faraday and Kerr effects which are measured with linearly polarized light, since linearly polarized light corresponds to a coherent superposition of equal “right” and “left” circular components. When viewed in a linear polarization basis, the preferential weakening of one of the circular components then leads to a polarization rotation.

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7 Semi-classical Response of Solids to Electromagnetic Fields

In the following we discuss the XMCD and Faraday effects through a phenomenological introduction of polarization dependent atomic scattering lengths or optical parameters.

7.11.1 Polarization Dependent Scattering Length and Optical Parameters Within the semi-classical framework the link of the experimentally observed effects to the fundamental atomic structure and the related electronic and spin structure remains hidden and is lumped into phenomenological parameters. A true understanding of all dichroism effects requires a quantum theoretical formalism, which will be developed in Chap. 11. We can, however, treat polarization dependent effects in our semi-empirical formalism by simply adding a polarization label p, that takes the values p = 0 for linear polarization, and p = ±1 for circular polarization, to both the x-ray scattering length and optical parameters that were linked in Sect. 7.5.5, so that f p = r0 Z + f  − i f  = p

p

 2π  p δ − iβ p . λ2 ρa

(7.113)

In particular, our previous expressions (7.52) and (7.53) then become λ2 ρa  p r0 Z + f  2π 2 ρa  p λ βp = f . 2π δp =

(7.114)

This formalism is particularly useful for the description of the interaction of circularly polarized light with ferromagnetic materials. The semi-classical description of the x-ray response of matter may be phrased in terms of phenomenological energy, ω, and polarization, p, dependent scattering lengths or optical parameters. Their values are based on experimental measurements of either the absorptive parts f  p or β p or refractive parts f  p or δ p (ω). Knowledge of one part allows the determination of the other by use of the Kramers-Kronig formalism.

7.11 Magnetic Dichroism and Faraday Rotation

373

7.11.2 Phenomenological Model For the quantitative derivation of magneto-optical effects one starts by considering x-rays that are purely circularly polarized since light of plus or minus angular momenta and a magnetic sample of plus or minus magnetization directions are both described by the same angular momentum basis states. We can conveniently utilize the description of light polarization in Sect. 3.2.6 and assume light propagation into the z direction in a Cartesian coordinate system (x, y, z). The unit polarization vectors are then defined according to (3.30) or 1 i  x = √ ( − −  + ) ,  y = √ ( − +  + ) , 2 2   1  1   + = − √  x + i y ,  − = √  x − i y . 2 2

(7.115)

They are illustrated in Fig. 7.22 and may be labeled by their angular momentum projection onto the z propagation direction (also see Fig. 3.4). Our definition directly maps onto the quantum description presented in Chap. 11, where the unit polarization vectors are expressed in terms of first order spherical tensors (see Table A.2). To emphasize the XMCD effect we assume that there is cylindrical symmetry about the incident beam direction in both charge and spin distributions, so that there are no other dichroic effects arising from different E orientations in the x−y plane.14 With this assumption we can distinguish the three cases of the incident light polarization  p through the labels p = 0, ±, where p = 0 corresponds to linearly or naturally polarized light with the E-vector somewhere in the x−y plane.

Unit polarization vectors photons

Unit magnetization vectors

Lz=

+

z +1

-

z

1

m

m x

x

z y

0

m y

mz=

z +1

z

1

z

0

Fig. 7.22 Definitions of unit polarization and magnetization vectors

If there was a preferential spin or charge orientation direction oˆ in the x−y plane, there would also be a linear dichroism effects for alignments of the polarization vector E  oˆ versus E ⊥ oˆ .

14

374

7 Semi-classical Response of Solids to Electromagnetic Fields

In Fig. 7.22 we also show three basic orientations of the unit magnetization vector ˆ in the same coordinate system. Similar to the photon angular momentum prom ˆ · ez onto the x-ray jections L z we also define magnetization projections m z = m propagation direction z. We then distinguish the three cases m z = 0, ±1. The value m z = 0 reflects a random magnetic orientation in the x−y plane or a non-magnetic sample. Of interest to us is how the incident polarized field is transmitted through a sample for different relative orientations of the x-ray polarization, described by unit vectors ˆ In the following we  p , and atomic magnetic moments, described by unit vectors m. shall phrase our phenomenological model in terms of phase or sign definitions that will directly map onto the quantum treatment in Chap. 11. For our assumed case of cylindrical magnetic and charge symmetry around the x-ray propagation direction z, we only need to distinguish three polarization notations. The case  0 denotes linear polarization or natural polarization with E somewhere in the x−y plane, and  + and  − , respectively, denote circular polarization with angular momentum (photon spin) in the +z and −z directions. We can then define polarization dependent optical parameters for the three polarizationcases by use of these polarization labels n˜ = 1 − δ 0 + i β 0 n˜ ± = 1 − δ ± + i β ± , δ ± = δ 0 ± δ , β ± = β 0 ± β .

(7.116)

We also have δ+ − δ− β+ − β− , δ = 2 2 + − + + β + δ− β δ , δ0 = . β0 = 2 2

β =

(7.117) (7.118)

Our definitions are illustrated in Fig. 7.23 through the energy dependent dichroic optical parameters for Fe and Co metals (also see Fig. 7.12 for Fe). Figure 7.24 shows a plot of the energy dependent polarization independent optical parameters δ 0 and β 0 defined in (7.118) and their dichroic differences δ and β defined in (7.117) for Fe, Co, and Ni metals. Table 7.2 lists the L3 resonant (peak) values for the optical parameters β 0 and δ 0 and their maximum dichroic magnetic contributions β and δ, corresponding to Fig. 7.24. The listed values should be taken to be approximate since they are sensitive to the experimental conditions, such as monochromator resolution, degree of circular polarization, magnetic alignment of the sample, or whether transmission or electron yield detection is used [72].

7.11 Magnetic Dichroism and Faraday Rotation

375 100

+

0.01

L3

Fe metal

80

0

-

0.005

60

L2

+

40

0

-

20

Optical constants

and

-20 -40

-0.005

-60 700

710 720 Photon energy (eV)

730

0.01 +

Co metal

80

0

-

0.006

60

f1 and f2 (Number of electrons)

0

0

+

40

0

-

20

0.002

0

0

-20

-0.002

-40 -0.006

770

780

790

800

Photon energy (eV) Fig. 7.23 Polarization and magnetization direction dependent optical parameters δ and β as a function of x-ray energy for Fe and Co metals. On the left we define the labels “+”, “–”, and “0” for the corresponding x-ray polarizations and magnetic directions. The main figure shows the energy dependent optical parameters β and δ for the three polarization labels for Fe and Co metal (left ordinate) and the related scattering factors f 1 and f 2 (right ordinate)

7.11.3 Transmission of Circularly Polarized X-Rays: XMCD For magnetic systems with the magnetization direction aligned parallel to the x-ray propagation direction it is convenient to work with circular basis states because they are eigenstates that are not changed upon transmission through the magnetically orientated sample. Only the field amplitude is changed.

376

7 Semi-classical Response of Solids to Electromagnetic Fields

Fig. 7.24 Optical parameters δ 0 and β 0 for Fe, Co, and Ni defined in (7.118) and their dichroic differences δ and β defined in (7.117) as a function of photon energy

8

L3

0

6

Fe L2

0

4 2 0 -2 -4 -6

Optical constants (x10 -3 )

8

700

710

720

730

Co

6 4 2 0 -2 -4 765

775

785

795

5

805

Ni

4 3 2 1 0 -1 -2 -3

850

860 870 Photon energy (eV)

880

In a circular polarization basis, the incident polarization is preserved upon transmission through a magnetic sample aligned along the x-ray propagation direction and only the amplitude of the field is reduced. For a given polarization direction p = ± for incident circularly polarized light and magnetization direction mˆ z , the transmitted fields are obtained by rewriting (7.23) by use of (7.117) and (7.118) as

7.11 Magnetic Dichroism and Faraday Rotation

377

Table 7.2 Optical parameters and their magnetic dichroism contributions at the L3 resonance positions of Fe, Co, and Ni metals, as shown in Fig. 7.24 ρa E0 λ0 β0 δ0 β δ 3 [atoms/nm ] [eV] [nm] Fe 84.9 Co 90.9 Ni 91.4

707 778 853

1.75 1.59 1.45

0.0084 0.0075 0.0050

−0.0060 −0.0042 −0.0028

0.0026 0.0019 0.0010

−0.0013 −0.0010 −0.0005

Listed are the atomic number densities ρa , the resonance energies E0 and wavelengths λ0 , the maximum values β 0 (on resonance) and δ 0 (below resonance) and the maximum dichroic contributions β (on resonance) and δ (below resonance), assuming propagation along the magnetization direction

Etrans = E 0 eikd e−(iδ+β)kd  p e− p mˆ z (iδ+β) kd . p

(7.119)

Here, δ = δ 0 and β = β 0 reflect polarization averages according to (7.118), and we assumed that for both polarizations the incident field amplitude has the same value E 0 . For linear polarization p = 0 or for mˆ z = 0 there is no magnetic effect. For circular polarization the transmitted intensity only depends on the whether p and mˆ z have the same or opposite signs. The transmitted intensity is obtained from the fields as p,mˆ p (7.120) Itransz = I0 e−2(β+ p mˆ z β)kd . The polarization and magnetization labels p and mˆ z with possible values ±1 enter equivalently. In our sign convention the transmitted intensity is smaller for p = mˆ z and larger for p = −mˆ z . The opposite is the case for the absorbed intensity since p,mˆ z

Iabs

p

p,mˆ

= I0 − Itransz .

(7.121)

When normalized to the same incident intensity (I0+ = I0− = I0 ), we define the XMCD difference intensity as p=mˆ

p=−mˆ z

IXMCD = Itrans z − Itrans

p=−mˆ z

= Iabs

p=mˆ z

− Iabs

.

(7.122)

For our sign convention we obtain IXMCD = −2I0 e−2βkd sinh[2β kd] .

(7.123)

The sign of the difference signal is therefore determined by the sign of β. Since β > 0 at the L−3-resonance of Fe, Co, and Ni, the dichroic contrast IXMCD is negative, in agreement with the usual sign convention [1].

378

7 Semi-classical Response of Solids to Electromagnetic Fields

7.11.4 Transmission of Linearly Polarized X-Rays: XMLD In order to see what happens to linearly polarized light upon transmission through ˆ parallel to the x-ray propagation direction, a sample with magnetization direction m we assume that the x-ray E-vector lies along an arbitrarily chosen x axis. The trick is that we can deduce the unknown transmission behavior of linearly polarized light in the sample from the known transmission of circularly polarized light discussed above. To do so we use the relationships between the unit polarization vectors  ± for circular and  x ,  y for linear light given by (7.115). For incident x-rays linearly polarized along x we can write the transmitted field in terms of two equal circular components given by (7.119) according to  1  + Etrans = √ E− trans −Etrans 2 & ' 1 0 0 = √ E 0 eikd e−(iδ +β )kd  − emˆ z (iδ+β)kd −  + e−mˆ z (iδ+β)kd . 2 (7.124) Note that the optical parameters are still defined in terms of the circular polarization response given by (7.117) and (7.118). The transmitted field has the form of an elliptically polarized wave of the general form (3.32). It can be rewritten in the linear polarized form (3.31) by use of (7.115) with the result 

  x cos[(δ − iβ)kd] +  y mˆ z sin[(δ − iβ)kd] . (7.125) The incident field, polarized along  x , is converted into an elliptically polarized transmitted field with complex x and y components. The first term due to the charge response preserves the polarization along x, while the second magnetic term, which depends on mˆ z , rotates it by 90◦ . The latter gives rise to the famous Faraday rotation discussed in more detail below. In contrast to (7.123), the transmitted intensity for incident linearly polarized x-rays is found to be Etrans = E 0 eikd e−(iδ

0

+β 0 )kd

Itrans = I0 e−2β

0

kd

.

(7.126)

It is independent of β, and there is no magnetic dichroism effect. If instead one measures the XAS spectrum for a sample of magnetization mˆ z by adding the intensities obtained with ± helicities, one obtains from (7.120) p=mˆ

Itrans =

p=−mˆ z

Itrans z + Itrans 2

= I0 e−2β

0

kd

cosh(2βkd)  I0 e−2β

0

kd

.

(7.127)

7.11 Magnetic Dichroism and Faraday Rotation

379

On the far right we have expanded cosh(2βkd) = 1 + 2(β)2 k 2 d 2 and only kept the first order term in β. This result is the same as (7.126).

7.11.5 Faraday Rotation The details of the x-ray Faraday rotation are best demonstrated by considering the transmitted field given by (7.125) for a thin film whose thickness is about one x-ray absorption length or d  20 nm. One may then utilize the fact that according to Fig. 7.24 the resonant dichroic parameters δ, β, which account for the magnetic response, are smaller than the non-dichroic parameters δ 0 , β 0 representing the charge response. This allows us to expand the cos and sin functions in (7.125) to first order in δ, β to obtain, Etrans  E 0 eikd e−(iδ

0

+β 0 )kd



 x +  y mˆ z (δ − iβ)kd



(7.128)

For a magnetic sample of orientation mˆ z = ±1 along the propagation direction z, the transmitted light has a real component along x and a complex component along y. Its effect on the polarization of the transmitted field may be nicely illustrated by considering the changes of the real, δ, and imaginary, β, dichroic contributions near a resonance. We consider the case of the L3 resonance of Co metal shown enlarged in Fig. 7.25a. 1 in Fig. 7.25a. Right before the resonance we have δ < 0, β  0, indicated by  Then (7.128) becomes Etrans  E 0 eikd e−(iδ

0

+β 0 )kd

   x +  y mˆ z δ kd .

(7.129)

The transmitted field has real x and y components so that the light remains linearly polarized, but the polarization vector has rotated by an angle ϕ, called the Faraday rotation angle, given by Faraday rotation : tan ϕ =

Ey = mˆ z kdδ . Ex

(7.130)

1 of β  0, δ < 0 under the On the bottom left in Fig. 7.25 we show the case  assumption mˆ z = −1. The negative values of both δ and mˆ z cause ϕ to be positive as shown in the figure. The other limiting case of (7.128) occurs at the resonance position where δ = 2 in Fig. 7.25a. The transmitted field becomes 0, β > 0, as indicated by  Etrans  E 0 eikd e−(iδ

0

+β 0 )kd



  x − i y mˆ z β kd .

(7.131)

In this case the y component is imaginary and the transmitted light has become elliptically polarized. The E-vector endpoint rotates on an ellipse with the rotation sense

380

7 Semi-classical Response of Solids to Electromagnetic Fields -3 x10

(a)

Co metal L3-edge

Optical constants

6 4

2 3

1

2 0

0 -2

0

-4

774

1

0

E

m

Ex z

rotated linear light

Emin

>0

y

x

E Ey

3

>0

y

x

sample

782

x

=0

E Emax

z

elliptical light

m

E Emin

Emax z

rotated elliptical light

Fig. 7.25 a Optical parameters δ 0 and β 0 and their magnetic dependences δ and β for the L3 resonance of Co metal. Below we show the transmitted field polarization for three photon energies 1 we have δ < 0, β  0. We show the polarization for mˆ z = −1 indicated in (a). At energy  2 we have δ = 0, β > 0, and we show the case mˆ z = +1 according given by (7.129). At energy  3 we have δ, β > 0 and we show the case for mˆ z = +1 described by to (7.131). For energy  (7.128)

defined by the sign of the imaginary component. The relative size of the principal axes of the ellipse define the ellipticity  according to Ellipticity : tan  =

|E y | E min = β kd . = |Ex | E max

(7.132)

The elliptical light may be described as having either a dominant linear or circular component. Owing to the imaginary y component, the Faraday rotation angle averages to zero over a full E-vector rotation (one wave cycle). In the lower middle of 2 where δ = 0, β > 0 and assume mˆ z = +1. The Fig. 7.25 we show the case , rotation sense of E is indicated by a black arrow, so that the transmitted light has a net angular momentum L z = −. 3 in (a) with β, δ > 0 on the lower Finally, we show the case indicated by  right of Fig. 7.25 under the assumption mˆ z = +1. This case is expressed by (7.128), and the transmitted light is elliptically polarized. The major axis of the ellipse is 1 due to the signs of δ rotated by a Faraday angle ϕ which is positive, as for case 

381

+

-

-2

=

(rad)

2

= +

-

Fe L2,3 edges

Twice Faraday rotation (deg)

(rad)

7.11 Magnetic Dichroism and Faraday Rotation

transform of + XMCD in absorption

Photon energy (eV) Fig. 7.26 Measurement of the photon energy dependence of δ + − δ − and β − − β + by Kortright and Kim [25] for Fe metal. a Faraday rotation measured through a 32 nm Fe film at 30◦ grazing incidence. On the left ordinate 2δ = δ + − δ − is given in radians and on the right the corresponding rotation angle in degrees. Note that it corresponds to twice the Faraday rotation angle. Solid circles are measured data-points, and the line is a smooth fit of the data by (7.130). b Experimental XMCD spectrum (blue curve) in comparison with the Kramers-Kronig transformed data in (a) (red curve)

and mˆ z . The rotation sense of E indicated by a black arrow results in a net angular momentum L z = −. The strongly energy dependent behavior of the dichroic optical parameters in conjunction with possible values mˆ z = ±1 results in a rich polarization dependence of the transmitted field illustrated in Fig. 7.25. The behavior has been studied experimentally for Fe, Co, and Ni metals by several groups [25, 73–75]. As an example, we show in Fig. 7.26 the measurement of 2δ = δ + − δ − and −2β = β − − β + for Fe metal by Kortright and Kim [25], which are seen to support the theory.

382

7 Semi-classical Response of Solids to Electromagnetic Fields

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Chapter 8

Classical Diffraction and Diffractive Imaging

8.1 Introduction and Chapter Overview In contrast to conventional light, x-rays have allowed us to see the invisible. Invisibility may be caused by the lack of penetration of visible light, preventing us to see below the outermost skin of matter. It also comes in the form of objects that are smaller than the wavelength of visible light, due to the diffraction limit. x-rays are best known because they can overcome both these limitations, prominently exploited over the last 100+ years in medical x-ray imaging and in x-ray crystallography. Today we still distinguish between the two different types of x-ray imaging which strongly overlap on the nanometer length scale [1]. Conventional x-ray imaging provides a real space picture of a sample with a spatial resolution limited by image forming lenses. Although the term “x-ray microscopy” is still used, today’s spatial resolution is well below the micrometer size. The strength of real space imaging is that the recorded image directly reflects the micro- or nanostructure of the sample. It is typically the preferred technique when the structures of interest are larger than about 100 nm, which is about 10 times larger than the state-of-the-art 10 nm resolution [2]. The present chapter starts with a short review of the development and capabilities of real space x-ray microscopy in Sect. 8.2. We discuss modern implementations of x-ray microscopes and emphasize the utilization of tuning the x-ray energy for atomic and chemical specificity and use of the polarization for the study of charge and spin-based orientational order on the nanoscale. Section 8.3 introduces diffractive imaging or lensless imaging where the limitation of x-ray lenses is overcome with the spatial resolution only limited by the wavelength itself, i.e. the diffraction limit. Now the recorded “images” are the reciprocal space diffraction patterns. Diffractive x-ray imaging is typically employed in the range below about 100 nm down to the atomic size of about 0.1 nm. The history of its theoretical formulation, based on the classical wave theory and the Huygens–Fresnel principle, is outlined with the key result being the Rayleigh-Sommerfeld diffraction formula. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_8

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In practice, approximations need to be made in applications of the RayleighSommerfeld formula, and in Sect. 8.4 we discuss consecutive simplifications that lead to the distinction of near-field Fresnel and far-field Fraunhofer diffraction. The Fraunhofer diffraction formula is particularly important since it links the field (not intensity) distributions in the source and detector planes through a Fourier transform. In Sect. 8.5 we touch upon one of the most profound scientific debates that continues to this day, the wave-particle ambiguity. The wave formulation of diffraction is still nearly exclusively taught in conjunction with the magical Huygens–Fresnel principle. This formulation, which allows light to go around corners, expressed by Sommerfeld as the deviation from rectilinear propagation, has been remarkably successful in explaining diffraction phenomena over a wavelength range from meters to sub-nanometers, covering more than ten orders of magnitude. The wave-particle ambiguity is briefly discussed by outlining the description of diffraction through Feynman’s probability amplitudes and de Broglie’s pilot wave theory. In Sect. 8.6 we discuss consequences of the Fraunhofer diffraction formula, such as the Rayleigh diffraction limit, the Arago-Fresnel-Poisson bright spot, and Babinet’s intriguing theorem. In Sect. 8.7 we formulate diffraction by matter in the Fraunhofer approximation. The atomic response is phrased in terms of polarization dependent optical parameters, introduced in Sect. 7.11.1, whose dichroic dependence is taken up to second order. Emphasis is placed on resonant effects at absorption edges that allow the detection of orientational order in nanoscale charge and magnetic domains and their distinction through polarization. The derived form of the diffraction pattern up to second order in the scattering factors or optical parameters will be utilized in last Sect. 8.11 where its close relation is shown to the modern Multi-wavelength Anomalous Diffraction or MAD technique. The remaining part of the chapter is devoted to how the reciprocal space diffraction pattern can be inverted into a real space image. We start by elucidating the origin of the famous “phase problem” in Sect. 8.8 through a simple example that clarifies the all-important effect of the phase in reconstruction of the real space image. Section 8.9 is devoted to the discussion of the most elegant phasing technique for 2D structures, Fourier transform holography (FTH). We show how the holographic intensity image can be directly inverted to obtain a real image of the object. We present an analytical solution of the phase problem for a double-slit object in the presence of a laterally displaced reference source and apply the technique for reconstructing the image of nanoscale magnetic domains. Section 8.10 gives an overview of non-holographic solutions to the phase problem. We review the historical development toward overcoming the phase problem in x-ray crystallography, which has certainly resulted in one of the great scientific accomplishment of the twentieth century, the elucidation of the atomic structure of matter. We only outline the physical principles of modern phasing techniques without discussing computer algorithms. The interested reader is referred to the recent comprehensive book by Jacobsen [1]. Crystallography relies on the presence of periodicity of the lattice in crystals or the periodicity of artificial stacks of macromolecules (see Fig. 1.14), resulting in

8.2 Real Space X-Ray Imaging

387

diffraction patterns dominated by sharp Bragg peaks [3–5]. We discuss the evolution of phasing from so-called direct methods, pioneered by Hauptman and Karle in the 1950s [6] to modern techniques. They depend on the increased scattering by heavy atoms that are either native or substituted, referred to as multiple isomorphous replacement or MIR. Of particular importance is the Multi-wavelength Anomalous Diffraction or MAD technique which exploits the change of the scattering phase at the heavy atom absorption edges. The section also discusses diffractive imaging of samples without long range periodic order. The Bragg peaks are then replaced by a so-called speckle pattern, and new phasing techniques are used which rely on computer programs with iterative algorithms that include specific boundary conditions also called “constraints” to guide the solution. We point out that in practice, all diffraction patterns are not sampled continuously but discretely because of the finite pixel size of detectors and discuss the requirement of sampling the diffraction patterns finely enough without loss of information. The section is concluded with a brief review of ptychography, which relies on sampling the diffraction pattern of an object through overlapping regions. The overlap provides a consistency check or constraint in the iterative phase retrieval. Section 8.11 discusses in more detail resonant phasing techniques related to MAD because they go the heart of x-ray matter interactions. They naturally evolved when tunable synchrotron radiation became available and gained additional momentum with the increase of coherent intensity at modern sources. We present the MAD formalism and extend it to the case of non-crystalline samples, emphasizing the beautiful interplay near resonances of the absorptive imaginary and phase-related real part of the atomic scattering factors or the related optical parameters.

8.2 Real Space X-Ray Imaging In the following section we briefly review real space x-ray microscopy techniques, with emphasis on the use of atomic resonances that provide elemental, chemical, and charge and spin sensitivity. The combination is often called x-ray spectro-microscopy which has been extensively covered in other books and review articles [1, 5, 7–9]. By now this technique is routinely used as evidenced by the nanoscale images presented in the Introduction of our book in Sect. 1.4.3. We will not cover the details of xray lenses and microscopes, limiting our discussions to the three main versions of spectro-microscopy used today. The rich nanoscale landscape of interest in x-ray imaging is illustrated in Fig. 8.1. It extends from the optical diffraction limit to the atomic length scale. The great advantage of real space x-ray imaging is that one can directly produce pictures of invisible objects that are smaller than the wavelength of visible light or the diffraction limit, indicated by the rainbow in Fig. 8.1. Polarization dependent x-ray microscopy also allows us to record images of spin-based structures that our eyes cannot distinguish even if they are larger than the diffraction limit. In some cases the nanostructures lack long range order like the “grains” in materials shown in the

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Characteristic Nanoscales in Matter -6

technology

spins

atoms

electrons

10 m 1 m Virus FM “bits” ~ 100 nm ~200 nm

100 nm

grains of materials ferromagn. vortex ~ 10 nm width ~ 10 nm

10 nm

transistor size ~ 50 nm DNA helix

C nanotube ~3 nm width ~ 2nm width

stable ferromagn. particles ~ 3nm

-9

10 m 1 nm atomic corral ~ 14 nm diameter

0.1 nm

spin density wave ~ 1 nm

charge stripes, orbital order ~ 1 nm

Fig. 8.1 Overview of structures of interest in the nanoworld, ranging from the optical diffraction limit (top) to the atomic size (bottom)

center of the figure. Also of interest are images of transient structures in response to a stimulation as shown in Fig. 1.22, or emerge as domains during fleeting moments in equilibrium, as in water [10]. Today, the observation of nanoscale dynamics down to the femtosecond timescale (see Fig. 1.12) is becoming increasingly important [8].

8.2.1 X-Ray Microscopes The spatial resolution of real space x-ray microscopy has significantly improved over the years as illustrated in Fig. 8.2, taken from a review by Kirz and Jacobsen [11], which outlines the historical development (also see [1]). As indicated in the figure, the development of photon-in/photon-out soft x-ray microscopy has been driven by the improvement in Fresnel zone plates (FZPs). In the hard x-ray range, the manufacturing of FZPs becomes more challenging since larger thicknesses are required due to increased transmission through the opaque regions. Their use has therefore been augmented by use of Kirkpatrick-Baez mirror pairs, multilayer diffractive optics, or compound refractive lenses. In the latter case one utilizes the fact that at x-ray energies the index of refraction is slightly less than unity and the refractive index δ is larger than the absorptive index β, as shown in Figs. 7.5 and 7.7. The optical glass lens in air is therefore replaced by an x-ray “air lens” incorporated into a low-Z material with low absorption [12]. More recently, multilayer lenses have been fabricated to create 2D focal spots below 10 nm [2].

8.2 Real Space X-Ray Imaging

Best resolution (nm)

1000

389

> 1 keV

< 1 keV

refractive lenses KB mirrors

100

FZP

Fresnel zone plate

10 Multilayer optics

1 1970

1980

1990

2000

2010

2020

Year

Fig. 8.2 Improvement of real space x-ray imaging methods with time [11]. For soft x-rays ( w/2.

(8.63)

We arrange two slits symmetrically around x = 0 and assume that they emit fields of real magnitude E 0 with flat-top distributions. The fields emerging from the slits of width w = a are given by E 1 (x − /2) = E 0 slita (x − /2) E 2 (x + /2) = E 0 slita (x + /2).

(8.64)

The fields E 1 and E 2 create the intensity diffraction pattern of our object. We now add a reference field that is created by a third narrower slit of width b  a that is offset from the center x = 0 by a distance ref and given by E 3 (x + ref ) = E ref slitb (x + ref ).

(8.65)

The total emitted field becomes E tot (x) = E 0 slita (x − /2) + E 0 slita (x + /2). + E ref slitb (x + ref ). (8.66) The diffracted field in a plane at large distance z 0 is given by the Fraunhofer diffraction formula (8.16) and consists of the Fourier transform of the total source field denoted F{E tot (x)}. By utilization of the linearity of the Fourier transform expressed by the Fourier addition theorem, Addition theorem : F{g + f } = F{g} + F{ f }

(8.67)

the holographic diffracted field is obtained as E(q) = E 0 F{slita (x − /2)} + E 0 F{slita (x + /2)} + E ref F{slitb (x + ref )}. (8.68) We now use the Fourier shift theorem in the form

8.9 Fourier Transform Holography: FTH

433

Shift theorem : F{g(x)} = G(qx ) ⇒ F{g(x ± /2)} = G(qx ) e±iqx /2

(8.69)

to obtain   E(q) = E 0 F{slita (x)} e−iq/2 + e+iq/2 + E ref F{slitb (x)} e−iqref .

(8.70)

Including the propagation factor from the source to the detector plane [see (8.16)], the Fourier transform is

qw  w F{slitw [x]} = √ sinc . (8.71) 2 λz 0 The holographic intensity pattern is given by

|E(q)| = 2

4E 02 

   

 2 a2 2 qa 2 q 2 b 2 qb sinc + E ref sinc cos λz 0 2 2 λz 0 2      object intensity pattern:o2 (q)

reference intensity pattern:r 2 (q)

 

qa  a b qb q + 2E 0 √ sinc 2E ref √ sinc cos cos[qref ] . 2 2 2 λz 0 λz 0       object field pattern:o(q)





reference field pattern:r (q)

(8.72) The first two intensity patterns identified by underbraces are also referred to as the autocorrelations of the fields, while the product of the two-field patterns in the second line is called the cross-correlation. The boxed part of the total expression holds the key to holographic recovery of the object in real space. Since it is the diffracted field of the object, its reverse Fourier transform into real space is the desired image of the object. All we have to do is figure out how to separate it from the rest. First, the Fourier addition theorem assures that the two autocorrelations functions and the cross-correlation function can be backtransformed separately. By choice of the distance ref > , this also provides a means to spatially separate the real space location of the backtransform of the all-important cross-correlation function from that of the two backtransformed autocorrelations. Before we illustrate this we need to point important aspect of the  out another backtransform of the entire hologram F −1 |E(q)|2 . It follows from the Fourier transform convolution theorem which for our real functions in (8.72) states as follows.

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Defining the convolution between two functions by the symbol , we can write the inverse Fourier transform of the product of two real functions, h(q)g(q), in terms of the Fourier convolution theorem F −1 [h(q) g(q)] = F −1 [h(q)]  F −1 [g(q)] = H (x)  G(x),

(8.73)

where the convolution is given by ∞ H (x)  G(x) =

    H x  G x −x  dx  .

(8.74)

−∞

This tells us that the Fourier backtransforms of both the autocorrelation and crosscorrelation intensities in (8.72) correspond to a convolution of the real space structures. There are three convolutions involved. Convolution of the two-slit object with itself will result in three conical peaks, just like the convolution of the two-circular holes with themselves, i.e. the Patterson map, in Fig. 8.19c. The single reference slit convolution with itself will be a single conical peak. Finally, the convolution of the reference slit with the two-slit object will result in two peaks which for b  a will reproduce the two-slit structure with a resolution limited by the finite size b of the reference slit. In the limit that the reference slit width approaches a Dirac δ-function, the object will be exactly reproduced.

8.9.1.1

Backtransform of the Hologram into Real Space

Because of the difficulty of evaluating the inverse transform of (8.72) analytically, we will illustrate it pictorially. Another reason for this approach is that in practice, holographic reconstruction always involves a pixelated image, typically recorded with a CCD detector. For our illustration, we choose our object as two slits of width a = 0.5 and separation  = 2, symmetrically arranged around x = 0 by ±/2. A narrower reference slit of width b = 0.05 is placed off-center at ref = 6, as illustrated in Fig. 8.20a. The slits act as sources with flat-top intensity distributions, as shown. The total holographic intensity pattern |E(q)|2 , given by (8.72), for E 0 = E ref = 1 and λz 0 = 1 is shown in Fig. 8.20b over a large range of −80 ≤ q ≤ 80, and its central part is shown in the inset enlarged over a range −10 ≤ q ≤ 10. To facilitate Fourier inversion, we put the holographic intensity distribution shown in Fig. 8.20b on a 1D grid of a large number of 32,000 pixels over the range −80 ≤ q ≤ 80. Because the diffraction pattern is in reciprocal space, the large q range assures optimum resolution, so that it is limited only by the size of the reference slit as discussed above. On the other hand, the pixel size determines the field of view of the backtransformed image, which is therefore very large for our small pixels. The

8.9 Fourier Transform Holography: FTH

435

Fig. 8.20 a Assumed two-slit object with separation  = 2 and width a = 0.5 plus a shifted reference slit at ref = 6 and width b = 0.05, acting as flat-top intensity sources. b Interference pattern or intensity hologram as a function of q given by (8.72), assuming E 0 = E ref = 1 and λz 0 = 1. The inset shows the enlarged central structure. c Fourier backtransform of the entire intensity hologram shown in black. It consists of a superposition of the backtransformed two autocorrelation intensities o2 (q) and r 2 (q) plus the backtransformed cross-correlation intensity o(q)r (q) in (8.72). The total backtransform is dominated by the three triangular peak Patterson map with an added small central spike due to the backtransformed reference intensity r 2 (q), shown in blue in the inset. The reconstructed two-slit object consists of mirror images that are displaced from the center by ref , highlighted by a red fill and shown enlarged in red in the inset

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8 Classical Diffraction and Diffractive Imaging

central part of total backtransformed image (about 1% of the total field of view) is shown in black in Fig. 8.20c. Our pedagogical simple slit example can readily be programmed to reveal these aspects.19 The backtransformed pattern in Fig. 8.20c looks confusing at first sight. We know, however, that it must have three additive contributions corresponding to the separate backtransforms of the two autocorrelation intensities o2 (q) and r 2 (q) of the object and reference, plus the backtransform of the cross-correlation intensity o(q)r (q) in (8.72). We can readily identify the separate contributions by individually backtransforming the parts o2 (q), r 2 (q), and o(q)r (q). In the inset of Fig. 8.20c, we show the two smallest contributions to the total black curve. The blue central spike is the backtransform of the reference autocorrelation intensity r 2 (q). The red curve is the backtransformed cross-correlation intensity o(q)r (q) of the object and the reference. The intensity scale of the inset shows that it is quite small relative to that of the total black curve which is dominated by three triangular peaks. They arise from the backtransform of the object autocorrelation intensity o2 (q) which is nothing but the Patterson map of the two-slit object, in complete analogy to that shown for the 2D two-hole case in Fig. 8.19d. The desired reconstruction of the object alone is reflected by the small displaced red-shaded rectangular areas, shown enlarged as a red curve in the inset. It consists of two mirror images, symmetrically positioned around the image center by the reference slit offset ref . The twin images arise from the fact that the cross-correlation term actually consists of two conjugate contributions. The choice of a reasonably large ref assures that the desired reconstructed twin images do not overlap with the autocorrelation structure in the center. The offset ref is why “off-axis” holography works so well. For our simple slit case,“in-line” holography works as well. If we place the reference slit right in-between the two object slits, i.e. ref = 0, with the same parameters as before we obtain the result shown in Fig. 8.21. The twin images of the object now overlap and appear together with the autocorrelation images in the center. Their separation from the autocorrelation structures is however a result of our simple 1D slit case. In general, one needs to use the off-axis holography approach.

8.9.2 Improved Reference Beams The extension from a single to multiple reference beams is illustrated in Fig. 8.22 [86, 87]. The figure shows how the offset distances and positions of the reference

19

Details of detector-limited sampling are discussed in W. F. Schlotter, Ph.D. dissertation, Chap. 2, Stanford University, 2007.

8.9 Fourier Transform Holography: FTH

437

Fig. 8.21 a Assumed two-slit arrangement as in Fig. 8.20a but with ref = 0. b Backtransform of the holographic diffraction pattern. In contrast to the pattern in Fig. 8.20c, the reconstructed two-slit object mirror images, identified by red shading, now overlap and occur in the center between the dominant three triangular Patterson map peaks

holes relative to the object, in form of an Ampelmännchen,20 need to be chosen to avoid overlap of the twin images for each reference hole with each other and the central autocorrelation images. The different object images can be added for improved signal-to-noise, which has been utilized for single-shot imaging of nanostructures with femtosecond XFEL pulses [88]. Another extension of off-axis non-iterative FTH is the replacement of the reference pinhole by a slit, whose fabrication by focused ion beam (FIB) milling is less sensitive to exposure time. One then utilizes emergence of the reference wave from the sharp edge of the slit or an L-shaped slit. The slit width determines the resolution in the direction perpendicular to the slit, and the sharpness of the terminal edge determines the resolution along the slit. The technique was suggested by Guizar- Sicairos and Fienup [89, 90] and termed holography with extended reference by autocorrelation linear differential operation (HERALDO). The differential holographic encoding technique was implemented for soft-ray imaging by Zhu and Guizar-Sicairos et al. [91, 92]. They compared the results obtained with the smallest reference pinhole of 40 nm that could be produced in 20

The shown Ampelmännchen figure was used on pedestrian crosswalks in East Germany before unification, as a green light signal to indicate “walk”. It was introduced to us by Andreas Scherz, a native Berliner.

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8 Classical Diffraction and Diffractive Imaging

Fig. 8.22 Demonstration of the use of multi-reference holes and choice of their positions to avoid overlap of the twin images for each reference hole with each other and the central autocorrelation images. From W. F. Schlotter, Ph.D. dissertation, Stanford University, 2007

a fabrication series with that for a 20 nm wide L-shaped slit, whose fabrication was less sensitive to the FIB milling process. By use of an L-shaped slit they achieved a resolution of 16 nm by synthesizing images in the Fourier domain from a single diffraction pattern. Direct comparison of the HERALDO results with images of the same sample obtained by diffractive imaging with iterative phase retrieval or real space microscopy with state-of-the-art zone-plate STXM and TXM microscopes showed a comparable or slightly improved resolution. We finish this section by stating three key attributes of Fourier transform holography. The reconstruction of the recorded hologram by Fourier inversion is fast, simple, and robust. The spatial resolution of the reconstructed image is determined by the reference hole size, if the maximum momentum transfer is kept large. The field of view of the reconstructed real space image is determined by the spatial frequency sampling of the hologram in reciprocal space, given by the pixel size of the CCD detector.

8.9.3 Application of FTH: Magnetic Domains The FTH formalism for magnetic domain diffraction is an extension of that discussed in Sect. 8.7.3 with addition of the reference hole contribution. We consider the circular polarization case treated in Sect. 8.7.3.1. The diffraction pattern (8.52) then contains additional self-interference and cross-interference terms due to the reference hole. The backtransformed diffraction pattern will consist of autocorrelation and crosscorrelation images of the reference hole, the central aperture, and the domain pattern inside. The Fourier transforms of the fields emerging from the three structures are separated through the addition theorem. The offset ref of the reference hole, assumed to be along x, is accounted for by the shift theorem, so that the total diffracted field is

8.9 Fourier Transform Holography: FTH

E diff (q) = E 0

439

eikz0 ikd −(iδ+β)kd e e D(q), iλz 0

(8.75)

where D(q) is the total polarization dependent diffracted field given by the Fourier transform of the transmitted field in the exit plane of the sample. By linearizing the exponential transmission (see (8.35)) D(q) depends on the relative alignment of the circular polarization helicity p = ± and the unit magnetization vector orientation mˆ z = ±1 along the x-ray propagation direction (see Sect. 7.11.3), and we have D(q) = e



−iqx ref



e reference

−i(qx x+q y y)

 dx dy +





D R (q)



− p mˆ z (iδ+β)kd] aperture



e−i(qx x+q y y) dx dy

aperture







DA (q)

mˆ z (x, y) e−i(qx x+q y y) dx dy . 

Dmˆ z (q)

(8.76)



The FTH intensity pattern is then given by |E diff (q)|2 which has auto- and crosscorrelation interference contributions. Our earlier pattern without the reference hole contribution (8.52) is then modified according to p

p

Idiff (q) =

I0

λ2 z 02

e−2βkd



 × |D R (q)|2 + |D A (q)|2 + |Dmˆ z (q)|2 (δ)2 +(β)2 k 2 d 2 −2 p mˆ z |D A (q)||Dmˆ z (q)|(δ sin φ +β cos φ)kd   +2|D R (q)| cos[qx ref ] |D A (q)| −2 p mˆ z |Dmˆ z (q)|(δ sin φ +β cos φ)kd .    cross-correlation for image reconstruction

(8.77) When this holographic diffraction pattern is transformed back into real space, the autocorrelation images due to the second line and the cross-correlation image arising from the third line will overlap in the center. The last underbraced term will create twin images, offset by ref from the center, which reflect the desired real space structure of the central aperture and the domain pattern inside it. The formalism was utilized for imaging the nanoscale magnetic domain structure of a magnetic multilayer by Eisebitt et al. [85]. As illustrated in Fig. 8.23, the incident circularly polarized x-rays from an undulator source were monochromatized by a spherical grating monochromator that determined the longitudinal coherence length. At the used photon energy 778 eV (wavelength 1.59 nm), corresponding to the Co L3 resonance, the longitudinal coherence length was 1.6 µm. The lateral coherence length was improved by spatial filtering of the beam from the monochromator by a

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8 Classical Diffraction and Diffractive Imaging

(a)

x-ray source

circular polarization

STXM image of magnetic domains

20 µm pinhole lateral coherence filter

1.5 µm

reference pinhole

aperture with sample

CCD holograms

Au mask SiNx membrane magnetic film with domains

SEM image

2 µm

left circular polarization

right circular polarization

(b) left circular polarization

right circular polarization

polarization difference

Fig. 8.23 a Experimental schematic. Monochromatized circular polarized x-rays are incident on a mask–sample structure after spatial coherence filtering. The object and reference beams are defined by the mask, and the resulting polarization dependent holograms, recorded with a CCD detector for p = ± at the Co L3 resonance, are shown in the lower right. The lower left inset shows the layered sample structure and an electron microscopy image. The upper right inset shows a STXM image of the magnetic worm domain structure within the sample aperture. b Fourier transforms of the holograms recorded with opposite circular polarizations and their difference. From [85] and W. F. Schlotter, Ph.D. dissertation, Stanford University, 2007

circular aperture of 20 µm diameter. The coherent central Airy disk of the transmitted beam (see Fig. 4.18) had a transverse coherence length of 9.1 µm at the sample position, where it illuminated a circular Au transmission mask of 1.5 µm diameter directly in front of the multilayer sample. The key element was the integrated reference, aperture plus sample design illustrated in Fig. 8.23a. It was fabricated by use of a 100-nm-thick Si3 N4 membrane on a Si support frame. The front side of the membrane contained a 600-nm-thick Au film, the other side a magnetic multilayer film of 50 Co(0.4 nm)/Pt(0.7 nm) bilayers on a 20-nm-thick Pt base layer. The multilayer which contained magnetic worm domains with magnetization directions perpendicular to the plane was capped with

8.10 Non-holographic Solutions to the Phase Problem

441

a 2 nm Pt layer to prevent corrosion. A focused ion beam was then used to cut the circular aperture out of the gold film, down to the Si3 N4 membrane. At a distance of 3 µm from the center of the aperture, a circular pinhole was drilled through the entire mask–sample structure by a focused ion beam. The high-aspect ratio pinhole had a conical shape along the beam direction with a minimum diameter of 100 nm at its exit. The diffraction pattern hologram is expressed by (8.77). It was recorded with an in-vacuum charge-coupled device (CCD) camera, with the photon energy tuned to the peak of the Co L3 resonance, where δ  0 and β  2 × 10−3 (see Fig. 7.25a). Since no beam stop was used, the central part of the holograms shown on the lower right of Fig. 8.23a was dominated by the Airy pattern from the circular aperture which had approximately 105 times the intensity of the outer part of the pattern. The Airy pattern is modulated by a high-frequency speckle pattern arising from the interference of the centrally transmitted field with that from the offset reference pinhole. Because of the limited dynamic range of the CCD camera, 50 image frames were acquired, each with a 10 s exposure time. The Fourier transform of the hologram is shown in Fig. 8.23b for opposite circular polarizations, with the two conjugate images, corresponding to the underbraced term in (8.77) separated from each other and offset from the central intensity arising from the other terms in (8.77). On circular polarization reversal of the incident soft xrays, the magnetic domain contrast inverts, in accordance with (8.77). It would also invert if for a given x-ray polarization the magnetic orientation of all domains was inverted. The polarization difference can be used to enhance the magnetic contrast and suppress any non-magnetic contributions as shown. Note that Babinet’s principle (see Sect. 8.6.3) does not apply to the magnetic domain case because the diffraction contrast is based on the relative directions of angular momentum vectors of x-rays and spins. The 10–90% contrast change in the 1D image profiles (not shown) had a width of only 50 nm, smaller than the minimum 100 nm diameter of the reference pinhole. It is likely that the conical shape of the pinhole, tapered because of the 8◦ convergence of the focused ion beam, acted as a capillary waveguide, and reduced the effective focal spot size that determines the spatial resolution.

8.10 Non-holographic Solutions to the Phase Problem Below we give a brief history of x-ray crystallography which elucidates how it was possible to overcome the phase problem in x-ray diffraction after the early work by von Laue, the Braggs, and others.

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8.10.1 Brief History of X-Ray Crystallography During the first 20 years of x-ray diffraction, relatively few crystal structures were solved. Solutions involved a “trial and error” approach based on Bragg’s insight that diffraction peaks arise from reflections from periodic lattice planes in crystals and Paul Ewald’s link of the “sphere of reflection” with the reciprocal lattice [93]. Experimental diffraction patterns of simple crystals with cubic lattices such as the alkali halides and diamond were compared with those calculated for envisioned structures, without the need for phase information. This trial and error method, however, rapidly stopped working for more complicated crystal structures. Remarkably, some of the modern solutions of phasing diffraction patterns were already envisioned in the late 1920s. As early as 1925, Mark and Szilard [94] recognized the importance of phase changes near absorption edges, which many years later led to the Single-wavelength Anomalous Diffraction (SAD) and Multi-wavelength Anomalous Diffraction (MAD) techniques. Another idea was the utilization of phase changes associated with the isomorphic substitution of lighter by heavier atoms first demonstrated in 1927 by Cork [95],21 which foreshadowed the development of Multiple Isomorphous Replacement (MIR). These ideas, however, were not fully developed at the time, and we will come back to them later. One of the key advances in x-ray crystallography came in 1934/35 when A. L. Patterson developed a method, now called the Patterson map, that serves as a starting point for solving crystal structures to this day [72, 73]. The Patterson map contains peaks at positions of all possible interatomic vectors between the atoms of a crystal. The inverse Fourier transform of the two pinhole intensity pattern, shown by the three peak structure in Fig. 8.19c, is a simple example of a Patterson map with the pinholes representing atomic sources. The map contains information on possible distances as well as directions of the diffracting holes. As is readily revealed by this simple example, additional information is required to truly solve the structure. One important aspect of the Patterson map is that it is dominated by the vectors between heavy atoms in a crystal. The combination of the Patterson map with the MIR technique proved especially valuable and was utilized by Max Perutz and coworkers in completing the structure determination of hemoglobin in 1954 [96]. Arguably the most important theoretical development of crystallography started in the 1950s, leading to so-called direct methods. The work was pioneered by Herbert A. Hauptman and Jerome Karle between 1950 and 1956 and was awarded the 1985 Nobel Prize in Chemistry [6]. The method is termed “direct” because it allows the structure determination directly from the data by comprehensive computer algorithms whose implementation became possible with the advent of increasingly powerful computers. An important conceptual contribution was made in 1952 by David Sayre [97], who considered the case where a crystal does not contain heavy atoms but only 21

Cork investigated the structure of crystals of a family called alums consisting of the symbolic chemical formula A·B·(SO4 )2 ·12H2 O, where “A” is a monovalent cation like K, Rb, Cs or Th and “B” a trivalent metallic ion like Al, Cr, or Fe. Cork also pointed out that S may be replaced by Se or Te, which later became an important use of MIR in protein crystallography.

8.10 Non-holographic Solutions to the Phase Problem

443

light atoms like C, N, O, and H. He then showed that one may utilize the peaked electron density at atomic positions to exploit the similarity between the electron density and its square. Any valid inversion of the diffraction pattern by direct methods has to give a density map on the smallest length scale of positive sign. Therefore any negative values do not reflect the real sample structure. On the largest length scale, the sample has a finite size and an inversion of the diffraction pattern should show no structure outside the sample boundary. These are the two most fundamental “constraints” on any structure solution. Today, the so-obtained complicated real space macromolecular structures may be visualized in different forms, ranging from early “ball and stick” models to the representation of larger structural motives by helix-shaped ribbons, as reviewed by Kaushik and Rath [98].

8.10.2 MIR, SAD, and MAD Image Reconstruction The MIR and MAD techniques, whose deep roots were mentioned above, combine chemistry-based (atomic modification) and physics-based (x-ray tuning) solutions to the phase problem. The history of MIR has been reviewed by Rossmann [69] and that of MAD by Ramaseshan [70] and Hendrickson [71]. From an x-ray point of view, the suggestion of MAD by Mark and Szilard [94] in 1925 brought together the fields of x-ray diffraction and x-ray spectroscopy. At the time, the change of x-ray response near resonances was known and referred to as “dispersion”, leading to the Kramers-Heisenberg dispersion formula [99] in 1925 and the Kramers-Kronig relations soon thereafter [100, 101]. The marriage of the two fields is reflected by the work of Hönl [102], who in 1933 used atomic wave functions to obtain oscillator strengths of x-ray absorption resonances that were then used to calculate dispersive atomic scattering factor corrections. The use of resonances for x-ray diffraction was further developed in the early 1950s by Bijvoet [103, 104] and Okaya et al. [105]. The two-wavelength MAD method was demonstrated in 1971 by Hoppe and Jakubowski [106] who determined the phase of Fe scatterers by use of two-photon energies below and above the Fe K-edge at 7,112 eV, corresponding to the characteristic Co Kα (6,930 eV) and Ni Kα (7,478 eV) emission lines. A similar approach was used in 1977 by Phillips et al. [107] who conveniently used tunable synchrotron radiation below and above the Fe K-edge. On the theory side, Karle [108] and Hendrickson [109–111] played an important role in the formulation of SAD and MAD. Because of its modern importance, we will discuss the theoretical formulation of MAD in more detail in Sect. 8.11. SAD experiments are a subset of MAD experiments. The small anomalous SAD signal from intrinsic S atoms was shown to be sufficient to phase the structure of the small protein crambin in 1981 by Hendrickson and Teeter [112]. Another milestone is the use of MAD to determine the structure by use of the Se K-edge in selenomethionyl proteins by Hendrickson et al. [110, 113].

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The heavy atom resonant scattering used in MAD may be envisioned as a holographic reference for the scattering by all atoms in the crystal. In practice, it utilizes the change of the phase of the heavy atom reference wave by recording diffraction patterns at different incident energies around the heavy atom absorption edge. The practical implementation of MAD was greatly aided by the availability of synchrotron radiation sources which not only provided higher intensity but readily allowed tuning of the x-ray energy to atom specific absorption edges. This has led to the dominant use of MAD for the determination of de novo crystal structures of biological macromolecules [71] (see Fig. 1.14).

8.10.3 Sampling of the Diffraction Pattern An important issue in all phasing techniques is that the diffraction pattern is recorded as a mosaic by use of a CCD detector with finite pixel size. This means that in practice, the inversion of diffraction patterns into a real space image involves discrete rather than continuous sampling by use of a discrete Fourier transform [114]. The pixel size has to be small enough to avoid signal overlap or aliasing. The latter is avoided by fulfilling the so-called Nyquist criterion. Nyquist’s paper of 1928 [115] considered when the information contained in a continuous temporal (telegraph) signal can be recovered when the signal is measured with a finite frequency bandwidth. In other words, when instead of the infinite number of sinusoidal Fourier components only a finite number are known. The work was extended by Kotelnikov in 1933 [116] and Shannon in 1949 [117], resulting in a theorem which today goes by various combinations of author names. The theorem appears to have been proven already by Whittaker in 1915 [118]. Nyquist stated his theorem by considering the sampling of a continuous temporal function of time, f (t), as follows. If the maximum cycle frequency of f (t) is B (which corresponds to the bandwidth in Hertz), the function f (t) is completely determined if it is sampled at temporal points that are 1/(2B) or less apart. In diffraction, the Nyquist criterion defines the maximum pixel size that can be used to record the continuous theoretical diffraction pattern without loss of information. It states that the maximum pixel size is determined by the inverse of twice the size of the diffracting object. Let us illustrate the powerful criterion by considering our earlier example of the holographic diffraction pattern of a two-slit object and a pinhole reference, discussed in Sect. 8.9.1. For convenience we show the geometry again in Fig. 8.24a. In our case, the highest frequency in the reciprocal space hologram is determined by the fringes produced by the object-reference interference term. It is the part of the total hologram pattern (8.72) given in normalized form by f (q) = 4 a b sinc

qa  2

 sinc

   qb q cos cos[qref ]. 2 2

(8.78)

8.10 Non-holographic Solutions to the Phase Problem

reference slits

0.8 Intensity

Intensity of interference term

1.0 (a) object and

0.6

b

a

0.4 0.2 ref

0.0 -4

-2

0.02 (c) enlarged 0.01

interference pattern

0 2 Distance

4

6

sampling rates: 0.45 Nyquist 0.5 0.4

0.00

-0.01 -2

-1 0 1 Momentum transfer q

0.02 (b) object-reference

2

interference pattern

0.01 0.00 -0.01 -0.02 -20

Intensity (arb. units)

Intensity of interference term

-6

445

-10 0 10 Momentum transfer q

20

(d) reconstructed object sampling rates 0.5 0.4

-2

-1

0 1 Distance

2

Fig. 8.24 a Object and reference geometry taken from Fig. 8.20. We have assumed a = 0.5,  = 2, b = 0.01, and ref = 6. b, c Interference pattern (8.78) on different q scales, with the maximum Nyquist sampling frequency [q]Ny = 0.45 given by (8.79) indicated in green in (c). d Backtransform of (8.78) for two sampling intervals, showing the reconstructed object, with only one of the twin images shown. The two-slit object is only reproduced when the sampling interval satisfies the Nyquist criterium q ≤ 0.45, as shown for the red reconstruction, where q = 0.4. For a larger sampling interval q = 0.5, the reconstructed blue pattern does not exhibit the proper separation between the slits ref

The highest frequency in q is determined by the modulation due to the two cos terms. It arises from the largest real space distance ref + /2 in Fig. 8.24a,22 which in our case is B = ref + /2 = 7. According to the Nyquist criterion, the reciprocal space pattern must be sampled at q intervals that satisfy q ≤ [q]Ny =

2π π = = 0.45. 2B 7

(8.79)

In practice, the pixels of our detector, located at a distance z 0 from the sample, must then have a size ρ ≤ [q]Ny z 0 /k. In Fig. 8.24d we show backtransforms of the intensity patterns in (b) given by (8.78) for two sampling intervals, yielding the blue and red reconstructed images of the object (only one of the twin images is shown). If the pattern is sampled at intervals q = 0.5 > [q]Ny , as indicated in blue in (c) and (d), the reconstructed pattern does not exhibit the proper separation between the slits ref due to aliasing. If the sampling interval is reduced slightly to q = 0.4 < [q]Ny as indicated in red 22

This is directly revealed by the Fourier transform of the two cos terms, which exhibits peaks at distances ref − /2, ref and ref + /2, as expected.

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8 Classical Diffraction and Diffractive Imaging

in (c) and (d), the object is properly reconstructed. The small change in sampling reveals the remarkable accuracy of the Nyquist criterion. In general, the case q < [q]Ny is referred to as “oversampling” while q > [q]Ny is called “undersampling”. A signal is said to be oversampled by a factor of N if it is sampled at N times smaller intervals than the Nyquist interval. Direct methods attempt to reconstruct the object image from the intensity of a single wavelength diffraction pattern. As shown in Fig. 8.21b for the simple two-slit case, the autocorrelation function or Patterson map extends to twice the size of the actual image. This means that if one wants to reconstruct the object starting with the autocorrelation, the pattern must be sampled with twice as many intervals as required by the Nyquist criterion. This led Bates in 1982 [119] to suggest the use of oversampling the diffraction pattern. The oversampling concept was extended by Sayre in 1991 [120]. Oversampling was demonstrated for non-periodic samples in 1999 by Miao et al. [121, 122]. They showed that sampling a diffraction pattern more finely than the Nyquist criterion corresponds to surrounding the electron density of the object with a no-density region. Finer sampling corresponds to an increase of the no-density region, and when its size exceeds the electron density region, sufficient information is recorded to retrieve the phase. This work emphasized the importance of the isolation of the object from the non-diffracting surrounding region when only a single pattern is recorded. In practice, oversampling is used today together with iterative phase retrieval algorithms [1, 123–126].

8.10.4 Ptychography The concept of ptychography was introduced by Hegerl and Hoppe in 1970 [127]. It exploits a convolution in reciprocal space between the Fourier transform of an entire object, F{O(r)}, with that of a finite part of the object, F{P(r − R)}, typically defined by a movable aperture. The name was coined by Hegerl and Hoppe by substituting the German word for convolution “falten” by the Greek word “π τ υξ =ptycho” (similar to Gabor’s substitution of “whole” by the Latin transliteration “holo” of the Greek word “oλo”). The name thus describes a solution to the phase problem that is essentially based on the Fourier convolution theorem. Remarkably, ptychography was nearly forgotten for almost four decades, mostly because of the limitation of coherent intensity emitted by electron and x-ray sources. It re-emerged through the work of Rodenburg and Faulkner in 2004 [128, 129] in conjunction with the use of iterative phasing algorithms. Owing to the increased coherent intensity of modern x-ray sources it was soon adopted by the synchrotron radiation community [130]. The history of ptychography within the broader efforts to overcome the phase problem has been reviewed by Rodenburg [131]. In his definition, ptychography is based on the use of at least two diffraction patterns, recorded with a relative lateral shift of object and illumination area. In practice, all that is required is a source

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447

of coherent radiation, an aperture, and a method of moving the aperture a known distance across the sample, which today is achieved by use of piezoelectric devices. We cannot adequately cover ptychography here and refer the interested reader to the recent book by Jacobsen [1]. We restrict ourselves to some important points. In its simplest form, an extended object is coherently illuminated and a moveable aperture defines a restricted spatial function P(r−R) that describes the transmitted field. By shifting the aperture to two or more positions, R, different regions of the object O(r) are sampled. Importantly, the aperture is moved so that there is overlap between the sampled regions. This way the diffraction pattern of the entire object can be obtained with the overlapping regions acting as a link or constraint that is incorporated into a suitable iterative phase retrieval algorithm. The overlap constraint helps with convergence problems in the phase retrieval, especially when the object is complex. For a single aperture position, Faulkner and Rodenberg’s ptychography algorithm [128, 129] reduces to the conventional phase retrieval one of Fienup [123]. From a physics point of view, ptychography exploits a convolution in reciprocal space between the Fourier transform of the object F{O(r)} and that of a finitearea function F{P(r − R)}, whose position is scanned across the bigger object. The diffracted intensity distribution has the form I (q) ∝ |F{P(r−R) O(r)}|2 = |F{P(r−R)}  F{O(r)}|2 .

(8.80)

On the right, we recognize the convolution theorem (8.73) which gave the technique its name. Ptychography derives the phase knowledge by recording two or more diffraction patterns which are related by the convolution theorem. For a more detailed discussion of x-ray ptychography and its increasing number of applications, the reader is referred to a recent reviews by Pfeiffer [130]. The comprehensive book of Jacobsen [1] also discusses phase retrieval algorithms. In the following section we come back to the MAD technique with a formal treatment, owing to its great importance in protein structure determinations and its use in reconstructing the speckle patterns of non-periodic nanoscale objects.

8.11 Multiple-Wavelength Anomalous Diffraction—MAD The technique of multiple-wavelength anomalous diffraction (MAD) has revolutionized atomic scale structure determinations of macromolecules (see Fig. 1.14). The term “anomalous” refers to the fact that historically diffraction experiments were typically performed with fixed energy laboratory x-ray sources that induced only non-resonant Thomson scattering by the sample. When the x-ray energy was close to the absorption edge of certain atoms, the changed scattering was termed “anomalous” (see Sect. 7.5.1). Today, “anomalous” scattering is used synonymously with “resonant” scattering. Thomson scattering is the x-ray response of the entire atomic electron cloud, which is largely independent of photon energy, while the

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8 Classical Diffraction and Diffractive Imaging

MAD method utilizes intra-atomic transitions that give rise to strong atom-specific resonance behavior (see Fig. 7.14).

8.11.1 MAD of Macromolecular Crystals The difference in scattering contributions from non-resonant Thomson and resonant MAD scattering is schematically illustrated in Fig. 8.25a, taken from the work of Hendrickson [110]. In order to enhance the signal from the molecular units they are stacked together to form a periodic arrangement or crystal (see Fig. 8.5). Each molecule in a crystal might contain a few thousand light atoms (such as C, N, and O) and few additional heavy atoms which are either native to the molecule (like Fe in hemoglobin) or are artificially introduced by isomorphous replacement (like Se for S). The light atoms are not shown individually in Fig. 8.25a but are contained inside the shown macromolecular envelopes. Heavy atoms are shown as circles. When the photon energy is not tuned to any of the absorption edges of the atoms, all atoms create Thomson scattering which is enhanced by the crystal periodicity in certain directions, resulting in Bragg peaks, as shown in Fig. 8.5a. When the photon energy is tuned to the heavy atom absorption edge, enhanced resonance scattering occurs, as shown for the atomic scattering factors f  and f  of Se in Fig. 8.25b. This enhancement is symbolized by replacement of the open circles for weaker Thomson scattering in the left panel of Fig. 8.25a by solid circles in the middle panel. On the right the sum of all contributions is shown. The phasing of the diffraction pattern is facilitated by measurement of the total diffraction data at a few wavelengths (usually 3–5) around the absorption edge of the heavy atoms. From the set of measurements it is then possible to extract the structure of the heavy atoms in the molecular unit cell. The determination of the locations of these heavy atoms then serves as a reference for building up a model for the molecular units.

8.11.2 Formulation of MAD in Protein Cystallography In crystals, the measured diffraction intensity is proportional to the squared amplitude of the molecular structure factor  2 atoms     f j (q) exp(i q · r j ) , |F mol (q)|2 =   j 

(8.81)

8.11 Multiple-Wavelength Anomalous Diffraction—MAD

(a)

Thomson scattering

Resonant scattering

Total scattering

Se

C,N,O,H

(b)

Number of electrons

f ’’

Se (Z=34) K-edge

12,652 eV

f’

Fig. 8.25 a Schematic of scattering contributions from a macromolecular crystal, whose stacked molecular units consist of many light atoms such as C, N, O, and H, contained inside the macromolecular envelopes, plus heavy atoms like Se shown as circles. Conventional non-resonant Thomson scattering is created by all atoms in a molecule. Resonant scattering occurs when the photon energy is tuned to the absorption edge of the heavy atoms (see panel (b) below). The enhanced resonance scattering of the heavy atoms is symbolized by solid circles relative to the weaker Thomson scattering indicated by open circles. On the right the sum of all contributions is shown. b Illustration of the resonance enhancement of the atomic scattering factors f  and f  at the K-edge of Se in a crystal of thioredoxin selenomethionyl thioredoxin from Escherichia coli [110]

449

Wavelength

(A)

where q is the momentum transfer. Here we have denoted the molecular structure factor as F mol (q) which consists of the summed contributions from the individual atoms, denoted f j (q) and given in units of number of electrons per atom. In MAD, one utilizes the strong difference of the scattering contributions of heavy atoms, e.g. Fe or Se, which serve as resonant scatterers, and those from the lighter atoms, typically C, N, and O. For the heavy resonant scatterers, denoted with a subscript “H” for “heavy”, the momentum, q, and wavelength, λ, dependent atomic scattering factors can be decomposed into three parts as discussed in Sect. 7.5.1. We write the atomic scattering factors in units of number of electrons given by (7.35),

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8 Classical Diffraction and Diffractive Imaging

with the replacement F(Q, ω) → f (q, λ) in our present notation as23 f H (q, λ) = f H0 (q) + f H (λ) − i f H (λ).

(8.82)

Leaving out the superscript “mol” in the molecular structure factor in (8.81) for brevity, the heavy-atom contribution, denoted FH (q), consists of a sum over those atoms whose scattering vectors q coincide with the reciprocal sublattice vectors G in the stacked molecular crystal expressed by H   atoms ei G·r j . FH (G) = f H0 + f H (λ) − i f H (λ)

(8.83)

j

Here the q-dependence of the atomic form factors f H0 (q) has been absorbed into the G-dependent factor. The heavy atoms have a non-resonant, wavelength-independent Thomson, f H0 , as well as a resonant scattering, f H (λ) − i f H (λ), contribution. We then rewrite the heavy (H) atom structure factor (8.83) in the form FH (G) =

f H0 

H atoms

 e

j



FHT (G)

+ FHT (G)  

i G·r j

 f H (λ) − i f H (λ) , f0  H 

(8.84)

FHR (G)

where the subscripts “HT” and “HR” distinguish heavy atom Thomson and resonance contributions. Denoting the low-Z or light atoms as “L”, their part of the molecular structure factor consists only of Thomson scattering, which occurs when the scattering vector q coincides with a reciprocal sublattice vector G associated with any of those atoms. With our labeling it is written as FLT (G) = f L0

L atoms

ei G·rn .

(8.85)

n

We can now group the light and heavy atom contributions by distinguishing Thomson and resonant scattering. The total Thomson contribution from the light and heavy atoms is given by FHT (G) + FLT (G) or Thomson : FT (G) = |FT | eiφT = |FHT + FLT | eiφT .

(8.86)

In conventional crystallography, only this Thomson scattering contribution, which through interference creates Bragg peaks, is measured. Even the Thomson contri23

The present notation differs from that in Sect. 6.2.4, where we used a capital letter, F(q) for the atomic scattering or form factor in units of [number of electrons], while a small letter f was used for the scattering length f (q) = r0 F(q).

8.11 Multiple-Wavelength Anomalous Diffraction—MAD

451

bution which scales with the atomic number Z is dominated by the heavy atoms. For a heavy atom like Se (Z = 34), the diffracted intensity |FHT |2 is enhanced over that of C (Z = 6) |FLT |2 by about a factor of 30. The reconstruction of the Thomson diffraction pattern typically starts with a Patterson map which emphasizes the Se positions and then proceeds by computationally intensive iterative phase retrieval, as in the direct method developed by Hauptman and Karle [6]. The MAD phasing formalism specifically hones in on the positions of the heavy atoms through their resonant contributions. By changing the wavelength one can now record several diffraction patterns whose difference is utilized in the phasing process. The resonant molecular structure factor of the heavy atoms is obtained from (8.84) as  Resonant : FR (G) = |FHR |eiφR = |FHT |

 f H (λ) − i f H (λ) iφR e . f H0

(8.87)

The total molecular structure factor is given by the sum of (8.86) and (8.87) as  Ftot (G) = |FT | e

iφT

+ |FHT |

 f H (λ) − i f H (λ) iφR e . f H0

(8.88)

The diffracted intensity is proportional to |Ftot (G)|2 . Leaving out the explicit Gdependence of the structure factors for brevity, the detected intensity can be written in the following parameterized form [5, 71, 111]. The diffracted MAD intensity consists of a total Thomson (denoted T) molecular structure factor contribution from the light (L) and heavy (H) atoms FT (G), given by (8.86). The Thomson contribution is phase shifted by φT − φR relative to the resonant contribution from the heavy atoms FHR (G). The latter is linked by (8.87) to |FHT | which appears in the MAD intensity expression used for structure determinations. The total MAD pattern is given by Idiff (G) ∝ |Ftot (G)|2 = |FT |2 + a(λ) |FHT |2 + b(λ) |FT | |FHT | cos[φT − φR ] + c(λ) |FT | |FHT | sin[φT − φR ].

(8.89)

The wavelength-dependent parameters  a(λ) =

f H (λ)

2

 2 + f H (λ) 2 f H (λ) 2 f  (λ) , b(λ) = , c(λ) = − H 0 (8.90)  0 2 0 fH fH fH

are entirely determined by the response of the heavy atoms. They can be determined in a separate x-ray absorption measurement of the sample which yields

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8 Classical Diffraction and Diffractive Imaging

f H (λ), supplemented by a Kramers-Kronig determination of f H (λ) (Sect. 6.6). The wavelength-independent Thomson response, f H0 , is available in tabulated form [132–134]. One is left with three unknowns, |FT |, |FHT | and φT − φR , which are determined from diffraction patterns recorded at three or more wavelengths near the heavy atom resonance (see Fig. 8.25). The molecular structure factor (8.88) contains important symmetry information that allows one to distinguish whether a system satisfies Friedel’s rule, |F(G)|2 = |F(−G)|2 or not. This arises from the equivalence of the transformations G → −G and [φT − φR ] → −[φT − φR ], so that the symmetry is directly reflected by the even cos and odd sin terms in (8.89) [5].

8.11.3 MAD Imaging of Non-crystalline Samples The extension of MAD to non-periodic nanostructures was treated and demonstrated by Scherz et al. [135]. The key difference between the diffraction pattern of crystals and non-periodic objects is that for crystals the pattern consists of well-defined bright spots, while for non-periodic objects the patterns is much weaker and smeared out, typically consisting of intensity speckles, as illustrated in Fig. 8.5. The non-periodic charge or magnetic patterns are given by the formalism developed in Sect. 8.7, which is closely related to the MAD formalism discussed above. In this section we show the close relationship between the MAD diffraction pattern for a macromolecular crystal, (8.89), and that of a film containing nanoscale charge domains given by (8.43). In order to see the correspondence we convert the optical parameter form in (8.43) to the scattering factor description in (8.89). The conversion utilizes the link established in Sect. 7.5.5. For dimensional clarification, the scattering factors are in units of [number of electrons per atom], which differ from the scattering lengths of dimension [length] simply by the single-electron Thomson scattering length r0 . In the following we will use the notation f  and f  for the resonant scattering factors. We can then convert the resonant optical parameters δ and β, defined in (8.33), which are multiplied by kd in (8.43) according to δ kd = f  r0 λρa d and β kd = f  C.   

(8.91)

C

Here ρa is the number density of atoms and the dimensionless number C = r0 λρa d reflects the number of atoms in the volume r0 λd. In addition to the differential domain transmission described by f  and f  we account for the averaged absorption of all domains by24 24

The corresponding parameter δ kd = f  C does not enter in the transmitted intensity.

8.11 Multiple-Wavelength Anomalous Diffraction—MAD

β kd = f  C.

453

(8.92)

By explicitly stating the phase shift in (8.43) as φ = φ A − φu , the charge diffraction pattern (8.43) takes the form film (q) = Idiff

   I0 −2C f  |DA (q)|2 + C 2 ( f  )2 + ( f  )2 |Duˆ (q)|2 e λ2 z 02 −2 f  C |DA (q)| |Duˆ (q)| sin[φ A − φu ]

 −2 f  C |DA (q)| |Duˆ (q)| cos[φ A − φu ] .

(8.93)

The factor in front of the curly braces is just an intensity scaling factor, with 1/(λ2 z 02 ) accounting for propagation and conservation of dimension since |DA | and |Duˆ | have  the dimension [area], while the average attenuation factor e−2C f is independent of the sample microstructure and simply scales the overall intensity. For direct comparison we write the MAD expression (8.89) by insertion of the parameters given by (8.90). Since Thomson scattering is constant, we then multiply the right side of this expression by f 0 , which simply scales the intensity. By also dropping the subscript “H” of the scattering factors we obtain  mol (q) ∝ Idiff



f0

2

  |FT (q)|2 + ( f  )2 + ( f  )2 |FHT (q)|2

+ 2 f  f 0 |FT (q)| |FHT (q)| cos[φT − φR ]

 − 2 f  f 0 |FT (q)| |FHT (q)| sin[φT − φR ] .

(8.94)

We can now compare the key structure dependent parts, identified by curly braces in (8.93) and (8.94). The obvious formal resemblance becomes more quantitative when remembering that in (8.93), |DA (q)| reflects the contribution from the atomless aperture hole, while the corresponding term f 0 |FT (q)| in (8.94) reflects non-resonant Thomson scattering. The two expressions thus differ in the nature of the non-resonant terms. Their relationship emerges by comparing the aperture contribution in Fig. 8.15 to the non-resonant Thomson contribution in 8.25a. Adding a non-resonant Thomson scattering background for the film, expressed by f 0 r0 λρa d = f 0 C in analogy to the resonant contribution (8.91), would result in the replacement DA (q) → f 0 C DA (q) in (8.93). We would then have  film (q) ∝ C 2 Idiff

  ( f 0 )2 |DA (q)|2 + ( f  )2 + ( f  )2 |Duˆ (q)|2 −2 f  f 0 |DA (q)| |Duˆ (q)| sin[φ A − φu ]

 −2 f  f 0 |DA (q)| |Duˆ (q)| cos[φ A − φu ] .

(8.95)

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8 Classical Diffraction and Diffractive Imaging

Since C 2 is a scaling factor, we factored it out of the curly braces to better reveal the close similarity with (8.94). There is, however, a remaining curious difference between the corresponding sin and cos phase terms and signs in (8.94) and (8.95). This is explained by our discussion of phase shifts upon resonant scattering in Sect. 7.5.4. While the resonantly scattered waves directly emerge from individual atoms in a molecule, they originate from the exit plane of the film after the incident wave has traversed it. As pointed out in Sect. 7.5.4, there is a π/2 phase shift between the atomic and film cases. This is also revealed by the factor −i in the Rayleigh-Sommerfeld diffraction formula (8.8), corresponding to a 90◦ phase shift. The difference in the sin and cos phase factors and signs then follows from the trigonometric identities ∓ sin[φ] = cos[φ ± π/2] and ± cos[φ] = sin[φ ± π/2]. Hence real space image reconstruction by MAD of molecular crystals with atomic resolution and of non-periodic domain structures with nanometer resolution proceeds similarly. In the following we will present an example of applying MAD for diffractive imaging of a non-periodic nanostructured sample.

8.11.4 Implementation of MAD for Non-periodic Samples We now apply the non-periodic MAD formalism by considering the experimental geometry and nanostructured sample shown in Fig. 8.26, studied by Scherz et al. [135]. We are interested in imaging polystyrene (PS) spheres of 90 and 300 nm in size deposited on a Si3 N4 membrane inside a field-of-view defining circular aperture of 1.2 µm diameter. For imaging of the PS spheres with an isotropic charge (bond) distribution, the x-ray polarization is unimportant. Of interest here is the reconstruction of the real sample image, shown by an SEM image on the top right, from the shown diffraction pattern recorded with a CCD detector. This is accomplished by use of the strong energy dependence of the optical parameters shown on the lower left. We note that the photon energy scale in Fig. 8.26 has been corrected to agree with that shown in Fig. 7.6 and the peak values of the optical parameters are smaller since they were obtained by use of the same polystyrene spheres and with the same energy resolution used in the imaging experiment. The intensity of the interference pattern for the experimental configuration shown in Fig. 8.26 is given by (8.43) in the optical parameter formulation. It can be separated into non-resonant and resonant components as discussed in Sect. 8.11.3. We can therefore rewrite (8.43) in the notation used by Scherz et al. [135] by denoting the non-resonant (N) aperture pattern DA = DN and the resonant (R) pattern by Duˆ = DR . Similarly, we denote the q-dependent phase shift between the complex amplitudes DN = |DN | eiφN and DR = |DR | eiφR as φ = φN − φR . We further abbreviate by defining new energy dependent parameters δ˜ = δkd and β˜ = βkd, where δ and β are shown in Fig. 8.26 to obtain (8.43) in the form,

8.11 Multiple-Wavelength Anomalous Diffraction—MAD

455

Monochromatic coherent source

SEM image Sample

500nm

Optical parameters

0.004

285 e V

C K-edge Polystyrene

CCD pattern

0.002

0

-0.002 275

280 285 290 Photon energy (eV)

295

Fig. 8.26 Schematic experimental setup for MAD diffractive imaging. The sample consisted of polystyrene spheres, of 90 and 300 nm in size dispersed from aqueous solution on a Si3 N4 membrane, with the field of view defined by a circular 1.2 µm diameter aperture created by deposition of a 800 nm thick Au film on the back of the membrane. An SEM image of the sample is shown on the top right. It was illuminated with a x-ray beam from an undulator source that was made spatially coherent with an aperture (not shown) and a monochromator allowed tuning the photon energy around the C K-edge of polystyrene. MAD phasing exploits the energy dependent interference of the resonant exit wave (orange) with the non-resonant exit wave (light blue). The center portions of the interference patterns recorded with a CCD detector at a photon energy of 285 eV are shown on a logarithmic scale. The strong energy dependence of the optical parameters of polystyrene is shown on the lower left [135]

   I0 −2βkd Idiff (q) = 2 2 e |DN (q)|2 + |DR (q)|2 δ˜2 + β˜ 2 λ z0   ˜ ˜ −2 |DN (q)||DR (q)| δ sin[φ] + β cos[φ] . (8.96) Here β is an energy independent absorption parameter that accounts for the transmission loss due to the sample support (e.g. a Si3 N4 membrane). To illustrate the distinct change of the diffraction pattern when the photon energy is changed from below to the resonance energy, we show in Fig. 8.27, simulated and measured diffraction patterns for the sample structure in Fig. 8.26. Below the resonance the patterns are the usual Airy pattern of a circular aperture which near the PS C K-shell resonance becomes modulated by diffraction from the PS spheres and, importantly, the symmetry is changed from even to odd centrosymmetric. From the energy scale in Fig. 8.26 it is seen that the photon energy of 285 eV coincides with

456 Fig. 8.27 On top we show the simulated central part of the diffraction pattern of the object in Fig. 8.26 at the two listed energies, calculated with (8.96). Underneath we show measured diffraction patterns at two different energies, for the central (green) and extended (blue) q range. Note the change from the mostly circular aperture pattern recorded below the resonance to a speckled pattern near the resonance. From D. Zhu, Ph.D. dissertation, Stanford University, 2010, and [135]

8 Classical Diffraction and Diffractive Imaging

Simulation 276 eV

286 eV

283 eV

Experiment 285 eV

283 eV

285 eV

the resonance in δ and the pattern is therefore largely due to a phase contrast (see Fig. 8.30 below). From knowledge of the optical parameters, we can derive analytical expressions for the three unknown parameters |DN |, |DR | and φ under the assumption that the sample is very thin. If at a selected energy both δ˜ = δkd  1 and β˜ = βkd  1, we can neglect the quadratic term in the optical parameters and obtain the simpler expression   ˜ ˜ sin[φ] + β(ω) cos[φ] . Idiff (q, ω) ∝ |DN (q)|2 − 2|DN (q)| |DR (q)| δ(ω) (8.97) Here we have explicitly identified the q and ω dependent terms. An analytical solution can then be obtained from normalized intensity measurements at three dif˜ ferent photon energies with associated optical parameters denoted δ˜i = δ(ω i ) and

8.11 Multiple-Wavelength Anomalous Diffraction—MAD

457

˜ β˜i = β(ω i)   I1 = |DN |2 − 2|DN | |DR | δ˜1 sin[φ] + β˜1 cos[φ]   I2 = |DN |2 − 2|DN | |DR | δ˜2 sin[φ] + β˜2 cos[φ]   I3 = |DN |2 − 2|DN | |DR | δ˜3 sin[φ] + β˜3 cos[φ] .

(8.98)

This directly yields the analytical solutions for the three unknown parameters |DR |, |DN |, and φ |DR | =

|DN | =



|DN |2 − I1

2|DN | δ˜1 sin[φ] + β˜1 cos[φ]



(I1 δ˜2 − I2 δ˜1 ) sin[φ] + (I1 β˜2 − I2 β˜1 ) cos[φ] (δ˜2 − δ˜1 ) sin[φ] + (β˜2 − β˜1 ) cos[φ] 

φ = arctan

(8.99)

1/2

(I1 − I3 )(β˜2 − β˜1 ) − (I1 − I2 )(β˜3 − β˜1 ) . (I1 − I2 )(δ˜3 − δ˜1 ) − (I1 − I3 )(δ˜2 − δ˜1 )

(8.100)

(8.101)

Depending on the sample, one may take advantage of computationally separating even and odd centrosymmetric components of the recorded images, expressed by the cos and sin terms in (8.97). The symmetry change with photon energy for our sample is evident from the patterns in Fig. 8.27. This allows simplification of the reconstruction problem by use of patterns at only two energies near the resonance, as utilized by Scherz et al. [135]. In practice, the analytical solution to the reconstruction expressed by (8.99)– (8.101) or by the symmetry-based two-energy method is only approximate since the thin-sample approximation is typically not fulfilled. For example, at an energy of 285 eV (λ = 4.35 nm), the optical constants for PS have the values indicated by the vertical line in Fig. 8.26, or β  δ  0.002. For the thick PS spheres of d = 300 nm thickness this corresponds to β˜  δ˜ = 0.87 which does not justify neglect of the quadratic terms. One therefore uses the linear analytical solution as an input to iterative image reconstruction. Figure 8.28 demonstrates the rapid improvement of the reconstructed image with the number of iterations which ultimately reproduces the object with a resolution of about 25 nm, limited by the statistics of the diffraction patterns at the highest q values [135].

8 Classical Diffraction and Diffractive Imaging

Fig. 8.28 Convergence of the object image reconstruction, characterized by decreasing iteration error. The upper plot shows how five independent runs of the iterative reconstruction algorithm followed a very reproducible trajectory to convergence. Shown below are the reconstructed images of the sample in Fig. 8.26 at iterations 2, 5, and 10. From D. Zhu, Ph.D. dissertation, Stanford University, 2010, and [135]

Reconstruction of the object Iteration error

458

1 0.1 0.01

0.001

5

10

iteration 2

8.11.4.1

15

20

25

number of iterations

iteration 5

iteration 10

Reference-Guided Phase Retrieval

We briefly mention another phasing method, which from a FTH point of view overcomes the limited resolution due to the size of the reference source, and from a MAD point of view speeds up the phase retrieval. The technique, referred to as referenceguided phase retrieval (RPR), implemented by Zhu et al. [136], adds a reference hole covered with a film of resonant atoms to the geometry shown in Fig. 8.26. By tuning the photon energy across the resonance of the reference film, both the intensity and phase of the reference beam can then be changed. In particular, the transmission of the reference film which exponentially changes with β and film thickness can be changed by many orders of magnitude before and at the peak of the absorption resonance as illustrated in Fig. 8.29 for a 200-nm-thick Co film. The reference beam can then be either utilized or eliminated [136]. 1

Transmission of a 200nm Co film

10-1 Transmission

Fig. 8.29 Illustration of the transmission of a 200 nm Co film near the L3 resonance, which changes by about five orders of magnitude within a few eV

10-2 10-3 10-4 10

Co L3 resonance

-5

765

770 775 780 Photon energy (eV)

8.11 Multiple-Wavelength Anomalous Diffraction—MAD

459

When the diffraction pattern of the object and reference is recorded below the resonance, the reference film transmission is high and one obtains the conventional holographic diffraction pattern. At resonance, the reference contribution is essentially absent since the film has negligible transmission, and one obtains the diffraction pattern of the object alone. The two patterns are readily scaled by the known change in wavelength. The backtransform of the hologram recorded below the resonance already contains low-resolution twin images of the object. One can use these images in a reference-guided phase retrieval of the object pattern recorded on resonance. The reconstruction process is greatly speeded up [136].

8.11.5 Phase Contrast Imaging: Combining FTH and MAD We previously showed in Fig. 8.23 the application of FTH to magnetic domain imaging. In that case the incident energy was tuned to the peak of the Co L3 resonance, where the transmitted field is strongly attenuated by absorption. More generally, the diffracted intensity, given by (8.77), depends on both the dichroic differences β and δ, as previously shown in Fig. 7.25a for the transmitted intensity. In some cases it is advantageous, e.g. for XFEL imaging, to reduce absorption as much as possible since the energy transferred to the sample in the absorption process heats the electronic system, which may lead to damage (see Sect. 15.2 below). This is accomplished by maximizing the contrast that arises from δ and minimizing that of β. This so-called phase imaging was demonstrated by Scherz et al. [137]. The experimental arrangement was similar to that in Fig. 8.23 and the sample was a magnetic multilayer consisting of 125 Co(0.25 nm)/Pd(0.9 nm) bilayers which similarly contained magnetic worm domains with magnetization directions perpendicular to the plane. The striking results obtained by tuning the incident energy through the Co L3 absorption resonance are shown in Fig. 8.30. At a given photon energy, the shown real space images are obtained by first subtracting holograms recorded with opposite circular helicities. The phase contrast is particularly sensitive to properly centering the two holograms at q = 0, since this location differentiates between odd and even symmetry behavior of components of the total holographic pattern. This symmetry is revealed in (8.77) by the odd symmetry sin[φ] term associated with δ and the even symmetry cos[φ] term associated with β. Once the holograms for the two polarizations are properly centered, the difference is backtransformed into real space. The real part of the backtransform gives the absorption contrast shown on top in Fig. 8.30 and the imaginary part the phase contrast shown at the bottom. The reconstructed images are plotted on the same contrast scale. The observed behavior closely follows the energy dependence of β and δ, A β  0 shown in the center of Fig. 8.30. Before the absorption peak, at position , and δ is negative. There is almost no absorption but finite phase contrast. At position B β and δ have about the same magnitude but opposite signs. This causes the ,

460

Absorption contrast (

)

2.0

( x10-3 )

Fig. 8.30 Results of an experiment similar to that in Fig. 8.23, performed by tuning the incident energy through the Co L3 absorption resonance. Shown are the real (top row) and imaginary (bottom row) parts of the backtransformed difference between holograms recorded with opposite circular polarization as a function of photon energy across the Co L3 resonance [137]. The contrast follows the energy dependence of the dichroic differences δ and β

8 Classical Diffraction and Diffractive Imaging

Co L3

1.5 1.0 0.5 0 -0.5 -1.0 -1.5 776

A

B

D

C

777 779 778 Photon energy (eV)

Phase contrast (

780

)

C β has its maximum while δ is close image contrast to be inverted. At position , D β and δ have to zero, and the absorption contrast dominates. At position , comparable magnitudes and the same sign. The contrast is therefore the same and B not inverted, unlike case . As mentioned above, the great advantage of phase imaging is mitigation of radiation damage since for comparable image quality the radiation dose may be substantially reduced (about a factor of 10). As such, the technique should be of great value for radiation sensitive samples, such as biological specimens or polymers.

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133. C.T. Chantler, J. Phys. Chem. Ref. Data 29, 597 (2000) 134. https://henke.lbl.gov/optical_constants/ 135. A. Scherz, D. Zhu, R. Rick, W.F. Schlotter, S. Roy, J. Lüning, J. Stöhr, Phys. Rev. Lett. 101, 076101 (2008) 136. D. Zhu, B. Wu, R. Rick, J. Stöhr, A. Scherz, Opt. Lett. 34, 2604 (2009) 137. A. Scherz et al., Phys. Rev. B 76, 214410 (2007)

Part III

Quantum Theory of Weak Interactions

Chapter 9

Quantum Formulation of X-Ray Interactions with Matter

9.1 Introduction and Overview The previous three chapters, forming Part II of this book, described the interaction of x-rays with matter in terms of classical EM waves. The classical wave theory quantitatively yields the Thomson scattering cross section, and scattering and absorption are linked through the Kramers-Kronig formalism. It needs to be recognized, however, that the classical approach does not allow the ab initio calculation of resonant scattering and absorption. In particular, the Kramers-Kronig transformation between absorption and scattering relies on the availability of experimental data. This shortcoming is overcome by the use of quantum theory, which allows the calculation of x-ray scattering factors or optical constants from first principles. It also provides a deeper appreciation of the true photon nature of light within the complete framework of QED. This chapter introduces the quantum theoretical formulation of photon-matter interactions, constituting the first of five chapters in Part III of our book. The quantum approach has its roots in a foundational paper by Kramers and Heisenberg in 1925 [1], written just before the formulation of quantum theory. Two years later Dirac quantized the EM field and used first order perturbation theory to derive the quantum mechanical expression for x-ray absorption and emission [2], later called by Fermi “Golden rule number 2” [3]. Shortly thereafter Dirac rederived the Kramers-Heisenberg formula in the second order quantum formulation [4]. We shall here use this modern quantum version of the theory and, giving credit to its architects, refer to it as the KramersHeisenberg-Dirac (KHD) theory. The basic premise of the KHD theory is that the photon-matter interaction may be described as a weak perturbation of the equilibrium state of matter. This allows one to view photons as a weak probe that can provide information on the unperturbed ground state of matter. The validity of this “weak field” perturbation approach is not obvious since the energy of a single x-ray photon is comparable to intra-atomic binding energies, and after absorption of a photon the atom is left in a highly excited state. The key assumption underlying the KHD theory is that the interaction process © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_9

467

468

9 Quantum Formulation of X-Ray Interactions with Matter

may be assumed to be quasi-instantaneous or time independent, so that the influence of the temporal evolution of the excited electronic system can be neglected. The KHD theory has been remarkably successful in describing all x-ray experiments during the first one hundred years of x-ray science, including those at the brightest synchrotron radiation sources. We note upfront that the KHD theory will eventually break down with increasing photon density which leads to two fundamental effects. First, when there are enough photons to excite the majority of atoms in a sample, photons at the back end of a pulse may experience a change of the electronic system. Secondly, at high intensity there are necessarily more than a single photon in a mode. These photons are indistinguishable and can act together to cause non-linear effects. This case will be treated in the last three chapters forming Part IV of our book. The present chapter provides an introduction of the foundation and formulation of the KHD theory. We first derive the complete photon-matter interaction Hamiltonian and by estimating the relative size of different contributions reduce it to the two leading terms which are linear and quadratic, respectively, in the vector potential (or electric field). We next outline the assumptions underlying the quantum mechanical calculation of transition rates that are expressed by the key result of the chapter, the KHD formula. We then give a brief overview of how the first order part of the KHD formula is used to calculate transition rates for x-ray absorption, x-ray emission, and Thomson scattering. We finish the chapter by discussing the application of the second order part of the KHD formula for the description of spontaneous and stimulated resonant scattering and two-photon absorption and photoemission.

9.2 The Photon-Matter Interaction Hamiltonian In order to describe the interaction of x-rays with atomic matter by means of quantum mechanics we need to consider the relative size of the energies associated with the radiation itself and the atomic building blocks of matter, so that we can formulate an appropriate Hamiltonian that allows us to treat the interaction through a perturbation approach. Furthermore, we have already discussed in a semi-classical picture that processes such as absorption, emission, and scattering have different interaction strengths or cross sections. Quantum mechanically, they therefore need to be treated in different orders of perturbation. We start by considering the typical relative size of energies that will allow us to establish a hierarchy in perturbation order.

9.2.1 The Pauli Equation Including the EM Field In deriving the Hamiltonian that describes the interaction of the EM fields with the electron charge and spin we need to start with a quantum mechanical description

9.2 The Photon-Matter Interaction Hamiltonian

469

that goes beyond the basic Schrödinger equation which does not include spin. Our starting point will be the Pauli equation for an atom (labeled “at”) which incorporates the spin into the non-relativistic time-dependent Schrödinger equation [5] in terms of a sum of the electron (e) and spin (s) contributions, according to  2  e p ∂ψ(r, t) ∗ + Vat (r) + s · B ψ(r, t). = i ∂t 2m e me    Hat = Hate + Hats

(9.1)

The solution of the Pauli equation for the ground state of a magnetic sample is very complicated and only possible by making approximations for the atomic potential energy Vat (r), into which we have lumped many terms [5]. The atomic Pauli equation (9.1) can be generalized to include terms due to an external EM field, similar to the general Dirac equation. This involves adding the photon Hamiltonian Hrad given by (3.52) or Hrad =

 pk

1 ωk a †pk a pk + . 2

(9.2)

As discussed by Strange [5], another modification is necessary, which consists of the replacement of the usual momentum term p by p + eA. This produces the correct motion of a charged particle with momentum p in the presence of a time-dependent vector potential A = A(r, t) where p and A commute. The total Hamiltonian for the atom in an EM field becomes H=

e e2 2 e 1 2 p + p·A+ A + Vat (r) + s · B∗ + Hrad . 2m e me 2m e me

(9.3)

We now take a closer look at the spin dependent term e(s · B∗ )/m e .

9.2.2 Evaluation of the Spin Dependent Part of the Pauli Equation In our treatment of the spin dependent term we use of the general expression for the electric field (3.37) ∂A(r, t) (9.4) E(r, t) = −∇(r, t) − ∂t and write the term B∗ in the presence of both an atomic electrostatic potential and a radiation field as [6]

470

9 Quantum Formulation of X-Ray Interactions with Matter

B∗ = B +

1  ˙ . p × (∇ + A) 2 2m e c

(9.5)

With the same substitution p → p + eA as above we obtain 1  ˙ (p + eA) × (∇ + A) 2 2m e c e ˙ + 1 (p × ∇) .  ∇ ×A+ (A × A) 2 2m e c 2m c2  e   spin-orbit

B∗ = B +

(9.6)

In the last step we have used B = ∇ × A according to (3.40), and in the evaluation of ˙ and kept only those quadratic the cross product have omitted terms linear in A (and A) in A or independent of A. The reason for omitting the linear terms in A is that they ˙ As indicated, the last represent smaller scattering contributions than the term A × A. term gives rise to the spin-orbit interaction between the electrons in the atom given by e s · (p × ∇) = ξnl (r ) s · l. (9.7) Hso = 2m 2e c2 On the right we give the usual form of the spin-orbit interaction, where the expectation value ξnl = ξnl (r ) is the spin-orbit coupling constant. In the context of this book, we are particularly interested in the values of ξnl for the 2 p core and 3d valence shells of the later transition metal atoms, which are listed in Table 9.1 [7, 8]. In particular the 2 p core shell is spin-orbit split into 2 p1/2 (L2 ) and 2 p3/2 (L3 ) states separated by 3ξ2 p /2. The 3d valence shell spin-orbit interaction is of great importance in magnetic transition metal compounds, since it determines the orientation of the easy magnetization axis relative to the lattice (the magneto-crystalline anisotropy) [6].

Table 9.1 Spin-orbit coupling constants for the 2 p core and 3d valence shells of the transitions metal ions [8] Ion config. ξ2 p (eV) Ion config. ξ3d (meV) Cr2+ p 5 d 5 Mn2+ p 5 d 6 Fe2+ p 5 d 7 Co2+ p 5 d 8 Ni2+ p 5 d 9 Cu2+ p 5 d 9

5.67 6.85 8.20 9.75 11.51 13.23

Cr2+ d 4 Mn2+ d 5 Fe2+ d 6 Co2+ d 7 Ni2+ d 8 Cu2+ d 9

30 40 52 66 83 102

Note that for s = 1/2, the spin-orbit splitting between the levels j+ = s + l and j− = s − l in a shell of angular momentum l is given by the Landé interval rule E = j+ ξnl = (2l + 1)ξnl /2. The Cu values are from [9]

9.2 The Photon-Matter Interaction Hamiltonian

471

9.2.3 The Complete Interaction Hamiltonian We can now rewrite (9.3) in terms of atomic contributions, photon contributions, and atom-photon interactions as H=

p2 + Vat (r) + Hso + Hrad    2m  e   photons atom e e2 2 e e2  ˙ . + p·A+ A + s · (∇ × A) + s · (A × A) me 2m e me 2m 2e c2    atom-photon interaction

(9.8)

By underbrackets we have identified the pure atomic, pure photon, and mixed photonatomic part, with the latter constituting the complete electromagnetic interaction Hamiltonian. It contains electron and photon terms. The electron contributions arise from the charge1 q = − e, mass m e , momentum p or spin s, while the photon terms are completely described by the time-dependent vector potential A, given in operator form by (3.55). Let us put a box around our final result. The complete electromagnetic interaction Hamiltonian is given by Hint =

e2 2 e e e2  ˙ . (9.9) A + p·A + s · (∇ × A) + s · (A × A) 2m e me me 2m 2e c2             H1

H2

H3

H4

The terms H1 and H4 are quadratic in A, while H2 and H3 are linear in A. According to (3.56) the vector potential A(r, t) = Aab + Aem can cause absorption or photon destruction and emission or photon creation. If the Hamiltonian depends linearly on A, the interaction can either destroy or create photons, but not both. Hence it can be used to describe the independent processes of x-ray absorption and emission. If the Hamiltonian is proportional to A2 , it can both destroy and create photons, which corresponds to scattering. The question arises whether we need to consider all four interaction Hamiltonians, or whether some can be neglected. We expect that the spin dependent interactions are weaker as we have seen in comparison with Thomson charge and spin scattering in Sect. 6.2.3. Also, we have seen in Sects. 7.3.3 and 7.3.4 that the direct coupling of the B field with the spins can be neglected at optical and higher frequencies. Let us therefore take a look at the relative size of the interactions by estimating the expectation values of the Hamiltonians. 1

Our result agrees with that given in the literature [5, 10, 11], but some authors define the electron charge as q = e, while we define it as q = −e.

472

9 Quantum Formulation of X-Ray Interactions with Matter

9.2.4 Relative Size of the Interactions In order to estimate the relative size of the energies corresponding to the four interaction Hamiltonians, we phrase all expressions in terms of the amplitude of the incident electric field |E|. We simply write |A| = |E|/ω and |B| = |E|/c. Further we assume |s| = 1 which corresponds to a typical atomic magnetic moment of |m| = 2μB and express the expectation value of the momentum operator according to  p = m e ωr , where r  is the expectation value of the core orbital involved in the electronic transition. We then obtain e2 E2 2m e ω2 |H2 | = e r  |E| e |H3 | = |E| mec e2  E 2. |H4 | = 2m 2e c2 ω |H1 | =

(9.10)

The Hamiltonians H1 and H4 are quadratic in the field and one readily obtains |H4 | =

ω e2  e2 2 = E2 . E 2 2 2 2m e c ω 2m ω m c2  e  e

(9.11)

|H1 |

The two interactions correspond to Thomson scattering by charges, H1 , and by spins, H4 . Spin scattering contains an additional factor ω/(m e c2 ), which due to the large value of m e c2  511 keV is much weaker than charge scattering. Spin dependent x-ray Thomson scattering has been measured in a beautiful tour de force experiment for antiferromagnetic crystals by de Bergevin and Brunel [12, 13]. We shall neglect H4 in the following. Next we compare the Hamiltonians H2 and H3 which are linear in the field and responsible for driving excitations of atomic core electrons. We obtain |H3 | =

ω λ e E = e r  E . 2    m e c 2πr  mec

(9.12)

|H2 |

Again, we find the reduction factor ω/(m e c2 ) for the spin dependent Hamiltonian H3 , but there is also another dimensionless factor λ/(2πr ) that depends on the ratio of the wavelength relative to the effective size of the atomic core shell r . We can estimate this value by means of the electronic binding energy of the core shell. We use a hydrogenic model where the H 1s binding or Rydberg energy of EB = 13.6 eV

9.2 The Photon-Matter Interaction Hamiltonian L3 (2p3/2) binding energy

B

(eV)

300 400 500 600 700 800 900 1000

1x10

) < >

-2

B

2p core shell radius r (nm)

Fig. 9.1 Core radius r  of the 2 p3/2 core shell in the 3d transition metals, calculated in a hydrogenic model according to (9.14). We have plotted r  both as a function of Z (black curve, lower scale) and 2 p3/2 (L3 ) binding energy EB (red curve, upper scale)

473

8x10

6x10

-3

-3

20 21 22 23 24 25 26 27 28 29 30 Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn

Atomic number Z and element

and the Bohr radius a0 are linked according to2 EB =

m e e4 , 2 2 (4π ε0 )2

a02 =

4 (4π ε0 )2 . m 2e e4

(9.13)

This leads to the generalized relation r 2 =

2 . 2m e EB

(9.14)

We have plotted r  for the 2 p3/2 core shell of the important 3d transition metals in Fig. 9.1. The values are found to be within a factor of 2 of those obtained with atomic Hartree–Fock calculations plotted in Fig. 8–10 by Cowan [7]. They are also approximately equal to the radial 2 p3/2 → 3d radial matrix element R plotted in Fig. 12.11. We can now estimate the value of the desired factor in (9.12) by using the values ω = EB and similarly for λ and obtain for the 3d transition metals, ω λ  5 × 10−2 . m e c2 2π r 

(9.15)

Hence we will in the following also neglect the Hamiltonian H3 . We can state as follows.

2 The expression of the Rydberg energy in terms of the electron’s rest mass and charge and Planck’s constant was derived by Bohr in 1913 [14], constituting a seminal step in understanding the atomic structure through spectroscopy.

474

9 Quantum Formulation of X-Ray Interactions with Matter

The interaction between x-rays and matter can be described entirely by the Hamiltonian e2 2 e A + p ·A . (9.16) Hint = 2m e me       Thomson scattering

photoelectric transitions

It does not explicitly include the electron spin, but the coupling to the electronic charge still allows access to the spin through the spin-orbit coupling. This gives rise to the x-ray magnetic dichroism effects discussed in Sect. 7.9.

9.3 Perturbation Treatment of X-Ray Scattering and Absorption If the interaction of the EM field with an atomic sample, described by (9.9), was as strong as the electronic interactions within the sample itself, we would have to consider the terms in the interaction Hamiltonian on an equal footing with the atomic and solid state ones. Fortunately this is not the case, and in the following section we shall discuss the concept of perturbation theory, whose essence is the fundamental assumption that the interactions of the EM fields with charges and spins are “weak”.

9.3.1 On the Use of Time-Dependent Perturbation Theory When the total Hamiltonian describing a system depends on time, the concept of discrete energy levels and stationary wave functions must be modified. The essence of approximate methods for the description of electronic excitations is to assume that the system is mostly time independent and that the time evolution can be approximately treated by assuming that it is either (i) a small change due to the weakness of the interaction, (ii) slowly varying with time (adiabatic approximation), or (iii) changing over a very short time interval (sudden approximation). Method (i) is similar to the concepts used in time independent perturbation theory where the total Hamiltonian is written in two parts, H = H0 + H and the “smaller” part H is treated as a perturbation of H0 . Time-dependent perturbation theory makes the same Ansatz. Here the time dependence is entirely contained in the “weaker” part H which acts as a time-dependent perturbation of the stationary Hamiltonian H0 . Let us explore whether the interaction Hamiltonian Hint given by (9.16) fulfills the requirement that it is “weak” relative to the electronic ground state Hamiltonian. From consideration of the semi-classical interaction of an EM wave with charges and spins, discussed in the last chapter, it is intuitively clear why elastic Thomson

9.3 Perturbation Treatment of X-Ray Scattering and Absorption

475

Timescales of electron motion Incident x-ray (~1 keV)

In field e- wiggles in time

x

E x

e- orbits in time

= 4 x 10 -18 s

orb

= 5 x 10 -16 s

a0

Fig. 9.2 Simple picture of an x-ray scattering process by an electron in a hydrogen atom. In the scattering process the electron is wiggled at the frequency of the incident electric field. At hν = 1000 eV we have τx = 1/ν = 4 × 10−18 s, which is much shorter than the orbit time of the electron τorb = 5 × 10−16 s. Therefore the scattering process takes place over a small fraction of the electron’s orbit. Even for the most intense x-ray sources today the electric field of the EM wave is still relatively weak, with an amplitude of only |E| ∼ 250 V/m. This is many orders of magnitude less that the atomic electric field |E| ∼ 1011 V/m, and the electron is therefore barely distorted from its orbit

scattering by an atom can be viewed as a small perturbation. Let us consider a scattering process by an EM wave off an electron in an atom, as illustrated in Fig. 9.2. For simplicity we use a hydrogen atom which has an electron bound by about E B = 10 eV, orbiting with the Bohr radius of a0 = 0.5 × 10−10 m and an orbit time of about τorb = 5 × 10−16 s, estimated from the relation τorb E B = h = 4.136 × 10−15 eV s, connecting energy and time. At the site of the electron the electric field of the nucleus is |E| = e/(4π ε0 a02 ) ∼ 5 × 1011 V/m. In Sect. 3.2.4 we have discussed the typical size of fields found in today’s state-of-the-art x-ray beams. They were readily calculated from the emitted intensity at third generation synchrotron sources, given by the Poynting vector. At an x-ray energy of 1000 eV, corresponding to a wave period of τx = 4 × 10−18 s we obtained field strengths of |E| ∼ 250 V/m, which are about nine orders of magnitude weaker than atomic fields. This leads to the picture shown in Fig. 9.2, where the EM wave wiggles the electron with a frequency that is large relative to the orbit frequency and an amplitude Aosc = e|E|/(m e ω2 ) that is very small relative to the classical orbit radius. Hence the scattering process can be treated by perturbation theory, both, because it is weak and because it is fast. It is more difficult to give a simple intuitive picture why an x-ray absorption event can also be viewed as a small perturbation, since excitation or ejection of a photoelectron from an atom involves a rather large energy transfer. The reason for our intuitive difficulty is that x-ray absorption is a quantum mechanical process and our simple classical thinking breaks down. If we compare x-ray scattering and xray absorption from a quantum mechanical point of view, they are rather similar. In one case a photon comes in, scatters and goes out, in the other, a photon comes in, scatters, and an electron goes out. From this picture one would therefore expect that both cases can be described in the same perturbative fashion.

476 Fig. 9.3 Illustration of spatial and temporal dimensions associated with a 1 keV photon

9 Quantum Formulation of X-Ray Interactions with Matter

= 1.2 nm t = 4 as

~ 0.2 nm

t ~ 0.6 as

Most important is that the excitation probability per atom is quite small, and the x-ray absorption event occurs over a very short time interval. The small excitation probability per atom follows from the small value of the x-ray absorption cross section, which has the approximate area of a core shell and is therefore much smaller than the total area of an atom. The interaction time can be estimated from the time that it takes the photon to cross the diameter of the inner shell, or about 10−18 s or 1 as. A similar time estimate is obtained from the oscillation time of the EM field, considering that the field that drives the excitation is large only for a fraction of its cycle period as illustrated in Fig. 9.3. For a photon energy of 1 keV (λ =1.2 nm), the field is large over a time window of order t = λ/(2πc)  0.6 as. The interaction process is so fast that it is over before the other electrons in the atom can adjust to the change. The perturbation treatment of the atomic response to x-rays is largely based on the fact the interaction process with a single photon is so fast (< 1 as) that it is not influenced by all atomic processes that occur on longer timescales. Consideration of the relative speed of processes is a common concept used to make approximations in physics. For example, it is the basis for the Born–Oppenheimer approximation [15] that states that electronic motion (of order 10−18 –10−15 s) is so much faster than nuclear motion (of order 10−12 s) that the electronic changes can be treated by assuming constant nuclear positions. In photoemission spectroscopy one uses the so-called sudden approximation which allows a description of the excitation process that is one-electron in nature [16]. The excitation of the “active” electron is viewed as “sudden” relative to the relaxation times of the other “passive” electrons. Another manifestation of the “speed” approximation is encountered in the “probe-before-destroy” concept, proven so valuable in single-shot protein imaging experiments with very intense X-FEL pulses [17, 18]. If the pulse length is kept short enough (of order 10 fs), the image can be recorded before the atoms have time to move, which proceeds with the speed of sound ( 1 nm/ps).

9.4 Kramers-Heisenberg-Dirac Perturbation Theory

477

9.4 Kramers-Heisenberg-Dirac Perturbation Theory The x-ray absorption and x-ray scattering cross sections are both calculated by consideration of the time-dependent perturbation of the sample by the EM field, described by the Hamiltonian (9.16). The time-dependent EM field induces transitions between an initial state |i and final state | f  where both states contain an electronic and a photon part. In time-dependent perturbation theory the time evolution of the system from a time t0 in the ground state |i to a time t in a final state | f  can be envisioned to proceed in multiple steps. In the first step, the electronic and photon parts of the system are decoupled. The electronic ground state |a evolves in an exactly predictable manner under the influence of the time independent Hamiltonian H0 according to3 |a(t1 ) = e−i(t1 −t0 )H0 / |a(t0 ).

(9.19)

At a time t2 = τ , the interaction (second step) takes place, and under the perturbation Hint the system is taken from an electronic state |a(τ ) with energy E a to an electronic state |b(τ ) with energy E b . We can also write this transition in a more general form  f |Hint |i, where the states |i = |a|n 1  and | f  = |b|n 2  are composite electronic and photon states with photon numbers n 1 and n 2 , respectively. In the third step the electronic state |b evolves entirely predictable again, similar to (9.19), |b(t3 ) = e−i(t3 −τ )H0 / |b(τ ).

(9.20)

In the second order, the transition is made in two steps through an intermediate state m by a process  f |Hint |mm|Hint |i. In the third order there are two intermediate states with an evolution characterized by  f |Hint |ll|Hint | j j|Hint |i and so on. Each step represents a higher order of perturbation. The formalism is developed in most standard quantum mechanics books, and we will not repeat it here but simply state the result up to second order, which we will refer to as the Kramers-HeisenbergDirac formula.

This comes about as follows. Assume a time-dependent state |α that satisfies the time-dependent Schrödinger equation of a time independent Hamiltonian H 3

i This integrates to

d |α = H|α . dt

|α(t) = |α(0) e−iHt/ = |α(0) e−iE α t/ .

(9.17) (9.18)

478

9 Quantum Formulation of X-Ray Interactions with Matter

9.4.1 The Kramers-Heisenberg-Dirac Formula The Kramers-Heisenberg-Dirac formula constitutes the starting point for the quantitative calculation of electronic x-ray processes under the assumption of the Born– Oppenheimer approximation and can be summarized as follows. The Kramers-Heisenberg-Dirac formula gives the transition probability per unit time, Wi f , from a state i to a state f up to second order according to

2

  f |Hint |mm|Hint |i

 f |Hint |i +

ρ(E f ) δ(E f − Ei ).

Ei − Em m (9.21) The dimension of Wi f is [1/time]. The dimension of  is [energy×time], that of the absolute value squared expression is [energy2 ], and ρ(E f ) is the density per unit energy of dimension [1/energy]. The tricky Dirac δ-function conserves energy and is formally defined only through its unitless integrated property (see Appendix A.3). It does not enter in the overall dimension of (9.21). 2π Wi f = 

Hint is the interaction Hamiltonian, and the wave functions |i and | f  are products of electronic and photon states and the energies E j are sums of electronic and photon energies. The sum is over all possible intermediate states of energy Em . If the final states | f  for the first order term and the second order term are the same, the two terms can interfere. The first term in the sum is sometimes referred to as “Fermi’s golden rule” in reference to Fermi’s review of the quantum theory of radiation [3]. The second term gives the transition probability from |i to | f  via a range of intermediate states |m, which are also products of electronic and photon states. The sum is over all possible intermediate states of energy Em through which the system can pass in a virtual sense that does not require energy conservation until the final state is reached. Since the first and second order terms are summed before the absolute value is calculated, there can be interference. In practice, this occurs when the interaction Hamiltonian in the first and second terms is different (otherwise the first term dominates), and the final states | f , composed of the product of the electronic final state and the final photon state, are the same. This is encountered in a purely elastic scattering process that starts from the electronic and vibrational ground state, and both the first order elastic Thomson scattering process and the second order elastic resonant scattering process return the system back to the initial state [19]. The final two terms in (9.21) represent the density of final states per unit energy (dimension [1/energy]), and Dirac’s δ-function assures energy conservation. Its value and dimension are defined only through its integral over energy (see Appendix A.3), so that the transition rate Wi f represents the transition probability per unit time.

9.4 Kramers-Heisenberg-Dirac Perturbation Theory

479

The KHD formula assumes an excitation process that occurs within an ultrashort time window under neglect of the detailed temporal evolution. The quasiinstantaneous events may be envisioned to occur when the incident electric field that drives the interaction process is large, as shown in Fig. 9.3. As discussed earlier, the perturbation approach may be justified by the fact that even the largest interaction process, x-ray absorption, typically requires an excitation attempt by more than 100 photons before an excitation occurs, and that the excitation of the core electron is so fast that the other electrons have difficulty to respond. The dominant Thomson and photoelectric terms in the interaction Hamiltonian (9.9) given by H1 and H2 may be written in the alternative vector potential and field forms introduced in Sect. 3.3.3 e2 2 e A + p ·A 2m e m  e      Thomson Photoelectric e2 e e2 = E2 − i E 2 − e r·E. p·E = 2 2m e ω meω 2m e ω2

Hint =

(9.22)

In rewriting the expression we have replaced the vector potential given by 3.55 by the electric field operator given by (3.61), according to A = −iE/ω, and the momentum operator by the length operator according to p = m e dr/dt = m e ω r. This is an example of the common concept of operator equivalents in quantum mechanics and corresponds to replacing the Coulomb gauge by the Göppert-Mayer gauge [20, 21]. The replacement of the vector potential A by the electric field operator E is possible because the interaction of the EM H field with the spins in a sample is extremely weak, of order ω/(m e c2 ). This allowed us to neglect the smaller terms H3 and H4 in the Hamiltonian (9.9). The negligible contribution of the magnetic field H in the x-ray region may be understood in a physical picture by the requirement of angular momentum conservation. At frequencies beyond the THz range, any excitation of the spin system requires that the associated change in angular momentum can be transferred to another reservoir, which in solids is the lattice. With increasing photon energy, the required angular momentum transfer is increasingly bottlenecked. This situation has been discussed in Sect. 7.3.3 in terms of the refractive index, which at optical and higher frequencies is only determined by the electric permittivity ε˜ and no longer by the magnetic permeability μ. ˜ In practice, the KHD formula is evaluated by assuming an interaction Hamiltonian described by one of the terms in (9.22), where the operators A = A+ + A− or E = E+ + E− are written in terms of photon destruction (superscript +) and creation (superscript −) operators according to (3.56) or (3.61). The photon part of the product states |i and | f  is then evaluated to leave only matrix elements between electronic states. We will show how this is done in the following chapters. The present chapter is intended as an overview of how different x-ray/matter interactions are described by the first and second order terms in the KHD formula.

480

9 Quantum Formulation of X-Ray Interactions with Matter

9.5 Overview of First Order Processes The basic quantum mechanical processes described by the first order term in (9.21) are x-ray absorption, x-ray emission, and x-ray Thomson scattering, illustrated in Fig. 9.4. In the following we shall outline the underlying concepts and the special absorption and emission cases, where the excitation process is resonant. In this case the atom is not ionized by excitation of a core electron into continuum states, but the core electron is promoted into a bound state, where it remains trapped over its lifetime.

9.5.1 X-Ray Absorption The strongest x-ray interaction with matter is x-ray absorption, schematically shown in Fig. 9.4a. The process is described by the interaction Hamiltonian ∝ p ·A+ ∝ r·E+ in (9.22) which is linear in the vector potential or field. In a quantum picture, the incident electric field or vector potential acts on an atomic core electron, with energy transfer from an incident photon, which gets destroyed, to an atomic core electron, which gets excited. In the figure we have assumed that the incident photon energy is larger than the atomic ionization potential, so that the electron is liberated as a

First order interaction processes (a) X-ray Absorption

(b) X-ray Emission

(c) Thomson Scattering 1

continuum states

2

ionization potential

core state

Fig. 9.4 Illustration of first order quantum mechanical processes in the interaction of a photon with the electronic charge of an atom. The atom is depicted as having discrete energy levels consisting of filled electronic shells and empty Rydberg states close to the ionization potential (IP) or vacuum level and a continuum of empty states above the IP. Electrons are shown as filled circles, holes as open circles. X-ray absorption and emission are first order processes where an x-ray is absorbed (destroyed) or emitted (created), described by the action of the destruction operator A+ and creation operator A− (see (3.58)) in the photoelectric Hamiltonian ∝ p · A in (9.22). Non-resonant Thomson scattering is also a first order process but involves the interaction Hamiltonian ∝ A2 in (9.22)

9.5 Overview of First Order Processes

481

photoelectron. When at a given photon energy, the kinetic energy of the emitted photoelectron is analyzed by an electron spectrometer, the measured photoemission spectrum will exhibit peaks reflecting electrons excited from different core shells. The intensities of the peaks directly reflect the ionization cross sections of the core shells. X-ray absorption spectroscopy (XAS) measures a photon energy dependent cross section that is the sum of all photoemission cross sections of shells with energy less than the photon energy. It is important to realize that although the total absorption cross section in the x-ray range decreases with photon energy as shown in Fig. 6.11a, the shell-specific photoemission cross sections behave the opposite way [22]. As illustrated for Ne in Fig. 1.27, at x-ray energies that exceed the binding energies of the innermost 1s electrons, the interaction cross section is the largest for the innermost core shell and smallest for the valence shell. For x-rays, there is a fundamental distinction between absorption and scattering, which becomes somewhat blurred for atoms in the optical region. Since atoms often have ionization potentials (IPs) of order of 10 eV, optical photons ( 1 eV) cannot induce electronic excitations. “Absorption” of incident photons is then sometimes associated with scattering of photons out of the incident beam, resulting in an “absorption” loss in the forward direction [23]. While this loss mechanism also exists for x-rays, it is clearly distinguished conceptually and quantitatively. The x-ray absorption process always involves an irreversible conversion of some or all of the energy of incident photons into electrons, which are ejected either through the direct photoemission process or the indirect Auger process. Atomic ionization cross sections change smoothly with photon energy and are tabulated as atom-specific quantities [24]. The tabulation assumes step-like changes at absorption thresholds and ignores the existence of resonant-like structures near the IPs. Structures in the cross sections above the IPs are also ignored. They may arise from multi-electron excitations, and in molecular and condensed matter samples oscillations are present due to backscattering of photoelectrons from the nearest neighbors, referred to as “Extended X-ray Absorption Fine Structure” or EXAFS [25–27].

9.5.2 Resonant X-Ray Absorption In practice, the spectral regions over which the absorption cross section varies smoothly are interrupted by step-like increases at absorption “edges”, which often exhibit strong resonances as shown in Figs. 1.18 or 6.6. The fine structure at core shell absorption thresholds, referred to as “X-ray Absorption Near Edge Structure” (XANES) or “near edge x-ray absorption fine structure” (NEXAFS) [16], corresponds to excitation of core electrons to empty valence states. It is therefore strongly dependent on the bonding environment of the selected atoms as discussed in Sect. 1.4.2.

482

9 Quantum Formulation of X-Ray Interactions with Matter

X-ray interaction processes and cross sections 107

Resonant x-ray absorption

core shell

Resonant elastic scattering

Interaction cross sections (barn/atom)

continuum states valence states

Fe Absorption

106

Photoemission, Non-resonant x-ray absorption

105

L2,3 4

10

103

Scattering

K

102

Elastic Thomson scattering

101

E

1

100

1000

e-

10,000

Photon Energy (eV)

Fig. 9.5 Illustration of the fundamental absorption and scattering cross sections per atom (1 b = 10−28 m2 ) as a function of photon energy, obtained from measurements of Fe metal between resonances [24] and around the L-edge resonances [6, 28]. The K-edge does not exhibit a resonance [29] and is approximated by a step. The associated photon interaction processes are also illustrated

Non-resonant and resonant x-ray absorption processes are identified in the upper half of Fig. 9.5 by schematics that have been added to our earlier Fig. 7.14. The shown cross sections σ of dimension [area] are given in units of barn (b) where 1 b = 10−28 m2 . They are obtained from the KHD transition probability per unit time Wi f by normalization to the incident photon flux 0 , given by (3.121), according to, σ =

Wi f . 0

(9.23)

9.5.3 X-Ray Emission The quantum mechanical x-ray emission process is illustrated in Fig. 9.4b. As discussed in Chap. 1, x-ray emission spectroscopy (XES) has a long history dating back to Moseley’s 1913 discovery of the characteristic change of the emitted Kα and Kβ photon energies for different elements [30], illustrated in Fig. 1.8. Like x-ray absorption, the XES process is also a first order process, calculated by evaluation of the first term in the KHD formula (9.21), i.e. Fermi’s golden rule, with the interaction Hamiltonian ∝ p ·A− ∝ r·E− in (9.22).

9.5 Overview of First Order Processes

483

The first order description is possible because one does not specify how the core hole state, which is the initial state of the process, has been produced. Hence it describes different cases where core holes have been produced by the interaction of photons or charged particles with a sample. The quantum process describes the spontaneous creation of a photon out of the zero-point vacuum of virtual photons by use of the energy contained in the exited electronic state. In the x-ray region, the filling of the core hole always proceeds by two competing spontaneous processes, resulting in emission of either a photon or Auger electron. In the soft x-ray region, the Auger process dominates while at hard x-ray energies x-ray emission is more likely [31]. The first order transition rate calculated by use of the KHD formula reflects only the dipolar radiative decay probability resulting in a photon. The intrinsically linked Auger decay channel is usually included only in an ad hoc fashion by defining a total decay rate that consists of the sum of the calculated dipolar radiative rate plus an Auger rate. The latter is obtained from more complicated electronic structure calculations [32] or from compiled semi-empirical values [33, 34]. If Auger decay is included in the ab initio formulation of x-ray emission, the treatment requires a second order formalism as discussed in Sect. 12.2.5. When the excitation step is included, one speaks of resonant inelastic x-ray scattering (RIXS). This process is also described by a second order formalism as discussed in Chap. 13.

9.5.4 X-Ray Thomson Scattering The process shown in Fig. 9.4c represents non-resonant or Thomson scattering. More specifically, we distinguish elastic and inelastic Thomson scattering. In inelastic Thomson scattering, the scattered photon has a reduced energy, with the energy difference used for atomic core or valence excitations [35]. In quantum theory, Thomson scattering is also a first order process described by the first term of the KHD formula (9.21) but with the A2 (E 2 ) interaction Hamiltonian in (9.22). The Thomson scattering cross section is considerably weaker than the xray absorption cross section and exhibits only a weak photon energy dependence as shown in the lower half of Fig. 9.5. Since the Hamiltonian is proportional to A2 , the scattering process may be pictured as the destruction of a photon by interaction with atomic electrons and recreation of a photon. Thomson scattering involves no specific electronic shells and scales with the total number of electrons per atom, Z . At hard x-ray energies, where the momentum of the photon needs to be taken into account, the scattering direction is angle dependent, as expressed by the atomic form factor discussed in Sect. 6.2.4. In the soft x-ray region, Thomson scattering becomes independent of the scattering direction and photons are scattered into random directions. In quantum theory, the random scattering direction, described classically by an outgoing spherical wave, is a consequence of the photon recreation process out of the random modes of the zero-point quantum vacuum.

484

9 Quantum Formulation of X-Ray Interactions with Matter

The most important consequence of Thomson scattering is Bragg diffraction, which is just the coherent superposition of elastic Thomson scattering from atoms in a crystal. The strong variation of the scattering cross sections near absorption thresholds in the lower half of Fig. 9.5 was historically called “anomalous scattering” [36]. Today, it is typically called resonant scattering since it involves a resonant absorption process followed by an x-ray emission process. The full description of resonant scattering requires a second order formulation as will be discussed next.

9.6 Overview of Second Order Processes Second order x-ray processes are treated quantum mechanically by the second term in the Kramers-Heisenberg-Dirac formula (9.21). It involves the product of two matrix elements of the interaction operator p · A, where A = A+ + A− . Since the second order formula involves three states, the initial, intermediate, and final states, the KHD formula applies to a variety of electronic processes, which may proceed via real or virtual intermediate states |m. As the name implies, second order processes involve two photons. One distinguishes the response of the atomic system to incident noninteracting single photons and the joint action of two incident photons. Figure 9.6 illustrates three examples of second order interaction processes.

9.6.1 Spontaneous X-Ray Resonant Scattering In spontaneous x-ray resonant scattering, illustrated in Fig. 9.6a, the double matrix element  f |Hint |mm|Hint |i in (9.21) is evaluated with the photoelectric operator in (9.22), in the form  f |p · A− |mm|p · A+ |i. In this case one only considers the actions of individual non-interacting incident photons, so that the initial state |i contains a single photon |n = |1 and a core electron in state |a, i.e. |i = |1, a. In the first step, the core electron gets excited to an unfilled electronic state |c and the photon gets destroyed (destruction operator A+ ). The created intermediate state |m then consists of the product of an excited electronic state |c and a photon state |n = |0 that contains no real photons but only the quantum vacuum. In the second step, the electronic energy stored in the intermediate state |m = |0, c can be used to spontaneously create a photon (creation operator A− ) out of the quantum vacuum, corresponding to the decay of the state |m into the final state | f . The final state then contains a single photon in an electronic state |b, i.e. | f  = |1, b. If |b = |a, we have spontaneous elastic resonant scattering and the case |b  = |a describes spontaneous inelastic resonant scattering.

9.6 Overview of Second Order Processes

485

Second Order Interaction Processes (a) Spontaneous resonant scattering

(b) Stimulated resonant scattering

(c) 2-photon absorption & photoemission

continuum states

|f

ionization potential

|m

|m 2

1

2

2

2

|m

1

|i

|f

|i

|f

1

|i

Fig. 9.6 Illustration of three kinds of second order interaction processes. In all cases, the system evolves from an initial state |i to a final state | f  through an intermediate state |m, where all states are products of photon number states and electronic states. a In spontaneous x-ray resonant scattering an incident photon excites a core electron to a partially empty valence state. The created intermediate state |m contains no photon, but a photon is spontaneously created out of the zeropoint quantum vacuum during the decay of |m to the final state | f . b In stimulated x-ray resonant scattering the intermediate state contains at least one other incident photon. This photon, which may have a different energy from the absorbed photon, drives or “stimulates” the emission of a second photon during the decay of |m to the final state | f . The emitted two photons constitute a cloned pair. c In two-photon absorption or photoemission, one incident photon excites the atom to a virtual intermediate state |m and a second incident photon provides the energy to excite the atom further to a bound or continuum state | f . Underneath we give the corresponding double matrix elements in the KHD formula

9.6.2 Stimulated X-Ray Resonant Scattering In stimulated x-ray resonant scattering, illustrated schematically in Fig. 9.6b, the double matrix element in (9.21) has the same form as for spontaneous resonant scattering,  f |p · A− |mm|p · A+ |i. The key difference is that one considers the combined action of two incident photons. This case was first treated by Maria Göppert-Mayer in 1931 [21], who derived the expressions for two-photon absorption and photoemission and stimulated Rayleigh/Raman scattering, long before such processes could be observed. Such events are typically very rare because they require a high density of suitable photons in the same interaction volume of the sample. In the x-ray regime, it took the advent of XFELs to observe stimulated decays [37, 38]. As in spontaneous resonant scattering, a core electron gets excited in a first step to an unfilled electronic state |c and one of the incident photons in the number state |n gets destroyed (destruction operator A+ ). The created intermediate state |m then consists of the product of an excited electronic state |c and a photon state |n − 1 that contains one less photon.

486

9 Quantum Formulation of X-Ray Interactions with Matter

In the coupled second step, the intermediate state |m = |n − 1, c may decay through the action of one of the remaining n − 1 photons (creation operator A− ). The decay is said to be driven or stimulated. The final state | f  then contains n photons in an electronic state |b, i.e. | f  = |n, b. In the stimulated decay processes, the photon that drives the decay clones itself, so that two photons emerge with the same polarization, energy and phase. If |b = |a, we have stimulated elastic resonant scattering and the case |b  = |a describes stimulated inelastic resonant scattering.

9.6.3 Two-Photon Absorption and Photoemission The two-photon absorption and photoemission process, illustrated in Fig. 9.6c, corresponds to (9.21) with the double matrix element of the form  f |p · A+ |mm|p · A+ |i. The essence of two-photon absorption is that two photons have to act together, in what Göppert-Mayer called an “Elementarakt”, an elementary non-separable act. The detection of two-photon absorption in the x-ray regime became possible only by use of XFELs [39].

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

12. 13. 14. 15. 16. 17. 18. 19.

H.A. Kramers, W. Heisenberg, Z. Phys. 31, 681 (1925) P.A.M. Dirac, Proc. Roy. Soc. A 114, 243 (1927) E. Fermi, Rev. Mod. Phys. 4, 87 (1932) P.A.M. Dirac, Proc. Roy. Soc. London A 114, 710 (1927) P. Strange, Relativistic Quantum Mechanics—With Applications in Condensed Matter and Atomic Physics (Cambridge University Press, Cambridge, 1998) J. Stöhr, H.C. Siegmann, Magnetism: From Fundamentals to Nanoscale Dynamics (Springer, Heidelberg, 2006) R.D. Cowan, The Theory of Atomic Structure and Spectra (University of California Press, Berkeley, 1981) G. van der Laan, B. T. Thole, Phys. Rev. B 43, 13401 (1991) S. Stepanow, A. Mugarza, G. Ceballos, P. Moras, J.C. Cezar, C. Carbone, P. Gambardella, Phys. Rev. B 82, 014405 (2010) M. Blume, J. Appl. Phys. 57, 3615 (1985) M. Altarelli, Resonant x-ray scattering: a theoretical introduction, in Magnetism: A Synchrotron Radiation Approach. Springer Lecture Notes in Physics, vol. 697, ed. by E. Beaurepaire, H. Bulou, F. Scheurer, J.P. Kappler (Springer, Berlin, 2006), p. 201 F. de Bergevin, M. Brunel, Phys. Lett. A 39, 141 (1972) F. de Bergevin, M. Brunel, Acta Cryst. A 37, 314 (1981) N. Bohr, Phil. Mag. 26, 1 (1913) M. Born, J.R. Oppenheimer, Ann. Physik 389, 457 (1927) J. Stöhr, NEXAFS Spectroscopy (Springer, Heidelberg, 1992) R. Neutze, R. Wouts, D. van der Spoel, E. Weckert, J. Hajdu, Nature 406, 752 (2000) C. Bostedt, S. Boutet, D.M. Fritz, Z. Huang, H.J. Lee, H.T. Lemke, A. Robert, W.F. Schlotter, J.J. Turner, G.J. Williams, Rev. Mod. Phys. 88, 015007 (2016) A. Pietzsch, Y.P. Sun, F. Hennies, Z. Rinkevicius, H.O. Karlsson, T. Schmitt et al., Phys. Rev. Lett. 106, 153004 (2011)

References

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20. G. Grynberg, A. Aspect, C. Fabre, Introduction to Quantum Optics, From the Semi-Classical Approach to Quantized Light (Cambridge University Press, Cambridge, 2010) 21. M. Göppert-Mayer, Ann. Phys. (Leipzig) 401, 273 (1931) 22. J.J. Yeh, I. Lindau, At. Data Nucl. Data Tables 32, 1 (1985) 23. R. Loudon, The Quantum Theory of Light, 3rd edn. (Clarendon Press, Oxford, 2000) 24. B.L. Henke, E.M. Gullikson, J.C. Davis, At. Data Nucl. Data Tables 54, 181 (1993) 25. D.C. Koningsberger, E.R. Prins, X-Ray Absorption: Principles, Applications, Techniques of EXAFS, SEXAFS and XANES (Wiley, New York, 1988) 26. J.J. Rehr, R.C. Albers, Rev. Mod. Phys. 72, 621 (2000) 27. G. Bunker, Introduction to XAFS: A Practical Guide to X-Ray Absorption Fine Structure Spectroscopy (Cambridge University Press, Cambridge, 2010) 28. C.T. Chen, Y.U. Idzerda, H.J. Lin, N.V. Smith, G. Meigs, E. Chaban, G.H. Ho, E. Pellegrin, F. Sette, Phys. Rev. Lett. 75, 152 (1995) 29. S. Pizzini, A. Fontaine, E. Dartyge, C. Giorgetti, F. Baudelet, J.P. Kappler, P. Boher, F. Giron, Phys. Rev. B 50, 3779 (1994) 30. H.G.J. Moseley, Phil. Mag. 26, 1024 (1913) 31. A.C. Thompson et al., X-Ray Data Booklet, 3rd edn. (LBNL, Berkeley, 2009). Available at: http://xdb.lbl.gov 32. W. Bambynek, B. Crasemann, R.W. Fink, H.U. Freund, H. Mark, C.D. Swift, R.E. Price, P.V. Rao, Rev. Mod. Phys. 44, 716 (1972) 33. M.O. Krause, J. Phys. Chem. Ref. Data 8, 307 (1979) 34. J.H. Hubbell, P.N. Trehan, N. Singh, B. Chand, D. Mehta, M.L. Garg, R.R. Garg, S. Singh, S. Puri, J. Phys. Ref. Data 23, 339 (1994) 35. W. Schülke, Electron Dynamics by Inelastic X-Ray Scattering (Oxford University Press, Oxford, 2007) 36. J. Als-Nielsen, D. McMorrow, Elements of Modern X-Ray Physics, 2nd edn. (Wiley, New York, 2011) 37. N. Rohringer et al., Nature 481, 488 (2012) 38. B. Wu, T. Wang, C.E. Graves, D. Zhu, W.F. Schlotter, J.J. Turner, O. Hellwig, Z. Chen, H.A. Dürr, A. Scherz, J. Stöhr, Phys. Rev. Lett. 117, 027401 (2016) 39. J. Szlachetko et al.: Sci. Rep. 6, 33292 (2016)

Chapter 10

Quantum Theory of X-Ray Absorption Spectroscopy

10.1 Overview As the first application of the Kramers-Heisenberg-Dirac theory we discuss the quantitative calculation of the x-ray absorption rate and cross section. X-ray absorption spectroscopy (XAS) is so fundamental and important that we devote the next two chapters to its various aspects. We will emphasize its dependence on the properties of light, such as the photon energy and polarization, and its dependence on the charge and spin of the sample. For most of our discussion the quantum theory will be phrased in terms an intuitive independent electron model, where electrons are excited from core states to partially empty valence states or into the continuum. Multi-electron correlation effects, giving rise to multiplet splittings in the observed spectra, will only be covered conceptually and through selected examples. Our emphasis will be on the pronounced resonant fine structure near absorption thresholds, known as NEXAFS or XANES, and we will not cover the extended x-ray absorption fine structure (EXAFS) above absorption edges but refer the reader to specific texts on the topic [1–3].

10.2 Quantum Formulation of X-Ray Absorption Spectroscopy (XAS) The first order term in the KHD formula (9.21) with the interaction Hamiltonian written as Hint = −e r·E in (9.22) describes x-ray absorption and emission, schematically shown in Fig. 9.4 a, b. The transition rate is given by Wi f =

2π |  f |e r·E|i|2 δ(E f − Ei ) ρ(E f ). 

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_10

(10.1)

489

490

10 Quantum Theory of X-Ray Absorption Spectroscopy

The field operator is given by E = E+ + E− according to (3.61), with E+ ∝ a pk destroying photons in XAS and E− ∝ a†pk creating photons in XES. For the description of x-ray absorption one first writes the states |i and | f  in (10.1) as products of electronic and photon states. Before the interaction we have |i = |a|n pk  and after the interaction | f  = |b|n pk − 1, where |a and |b are electronic states. The matrix elements in the KHD formula can be factored into a photon and electronic part, and the photon part is evaluated as   n pk − 1|E+ |n pk 2 = n pk ω 20 V pk

(10.2)

The transition rate can then be expressed in terms of the electronic transition operator and wavefunctions as abs = Wa→b

 2 n pk c 4π 2 ω pk αf  b| r ·  p eik·r |a ρ(Eb ) δ[ω pk − (Eb −Ea )] . (10.3) V pk       σXAS  pk

Here  pk is the photon flux per mode with dimension [number/(area×time)], σXAS the absorption cross section of dimension [area],  p the unit photon polarization vector, Eb − Ea the energy difference between the electronic states, ρ(Eb ) the density of final states per unity energy of dimension [1/energy], and αf =

e2 1  4π 0 c 137

(10.4)

is the dimensionless fine structure constant in SI units which characterizes the strength of the photon-electron interaction. The expression (10.3) still contains the photon wavevector k whose magnitude k = 2π/λ characterizes the spatial extent λ of a wavecycle. The neglect of the kdependence underlies the famous dipole approximation eik·r = 1 + ik · r + ...  1.

(10.5)

It turns out that this approximation, which assumes that the atomic size reflected by r is small relative to the wavelength λ = 2π/|k|, is remarkably good for the description of almost all aspects of XAS. We will, however, see in Sect. 11.4 that the dipole approximation breaks down for the description of XNCD, since the k dependence is required for x-rays to couple to the spatial “twist” in the atomic charge density for chiral systems. In the dipole approximation we can write the absorption rate in the following form.

10.2 Quantum Formulation of X-Ray Absorption Spectroscopy (XAS)

491

The polarization ( p) dependent x-ray absorption cross section in the dipole approximation is given by  2 p σXAS = 4π 2 ω αf  b| r ·  p |a ρ(Eb ) δ[ω − (Eb −Ea )].

(10.6)

It depends on the fundamental dimensionless fine structure constant αf , which characterizes the strength of the photon-electron interaction and the photon energy ω, the transition probability given by the squared matrix element between electronic states |a and |b, and the electronic density of final states per unit energy ρ(Eb ) (dimension [1/energy]). The δ-function assures energy conservation. Spin degeneracy may be included in the states |a and |b or the density of final states.

10.2.1 Photon Flux, Intensity, and Absorption Cross Section The incident photon flux  can be written in different forms. In practice, an x-ray beam incident on a sample of thickness d consists of a total number of photons n 0 within an illuminated area A (see Fig. 3.2), where the photons may be in different modes. The photon flux through the sample volume V = Ad is then 0 = c

n0 . V

(10.7)

When the volume is reduced to the single-mode coherence volume given by (3.62) which contains only photons in the same mode pk, we have with V pk given by (3.106),  pk = c

n pk n pk c  pk = 3 , V pk λ ω

(10.8)

where  pk is the energy bandwidth. The flux can be pictured as the number of photons that flows with the speed of light through a specified volume according to =

n pk c n0c I0 2 0 c E 02 = . = = V pk V ω ω

(10.9)

We have also expressed the flux in terms of the incident intensity I0 with dimension [energy/area/time] and field amplitude E 0 [V/distance].

492

10 Quantum Theory of X-Ray Absorption Spectroscopy

10.3 Non-resonant Absorption: Excitation into Continuum States The cross-section (10.6) describes non-resonant as well as resonant absorption processes, which are distinguished by the description of the final electronic states b reached by the transition. In non-resonant processes the electron is excited into continuum states, corresponding to photoemission. The cross-section expression for continuum excitation has been derived in [4–7], and the specific atomic shell-dependent cross sections have been calculated and tabulated by Yeh and Lindau [8]. In the following we shall outline the cross-section derivation for this case and then consider the more interesting resonant cross sections, owing to their atomic, chemical, and spin dependent specificity.

10.3.1 Wavefunctions Wavefunctions for atomic spin orbitals may be written in the form [7, 9–11] n,l,m,m s = Rnl (r ) Yl,m m s ,

(10.10)

 where Rnl (r ) has the dimension [1/ length3 ] and satisfies the orthogonality condition and normalization ∞ Rnl (r )|Rn  l (r ) =

Rnl (r )Rn  l (r ) r 2 dr = δnn  .

(10.11)

0

We note that the radial functions, obtained by solving the Schrödinger equation, are sometimes defined as P(r ) = r R(r ) [9, 11]. The simplest analytical functions are modified functions for hydrogen where Z is taken to have the effective value Z ∗ < Z to mimic screening effects [9]. For the description of K (1s → 2 p) and L (2 p → 3d and 2 p → 4s) shell excitations, the most important hydrogenic wavefunctions are1

Z ∗ 3/2 −Z ∗ r/a0 R1s (r ) = 2 e a0 ∗ 5/2 Z r ∗ R2 p (r ) = √ e−Z r/2a0 2 6 a0 ∗ 7/2 4 r2 Z ∗ R3d (r ) = √ e−Z r/3a0 . 81 30 a0 1

(10.12)

These functions are those given by Cowan [9] in Table 3–1, converted to our notation. The ∞ normalization is readily evaluated by means of the integral 0 r m e−ar dr = m!/a m+1 .

10.3 Non-resonant Absorption: Excitation into Continuum States

where a0 = 4π 0

2 m e e2

493

(10.13)

is the Bohr radius a0 = 0.529 × 10−10 m. The key in the cross-section derivation is the description of the wavefunction of the photoelectron moving away from the atom. It is not limited to a finite volume and is influenced by the Coulomb interaction with the left-behind ion. The photoelectron wavefunction depends on the kinetic energy of the outgoing photoelectron, given in terms of the photoelectron wavevector q (with units [1/length]) by Ekin =

2 2 q . 2m e

(10.14)

In general, a free electron state can be described by a function that is constructed from either a plane wave or spherical wave basis.2 In analogy to the bound state wavefunction (10.10) we can write the continuum free electron wavefunction as q,l,m,m s = Rq,l (r ) Yl,m m s .

(10.16)

The radial part can be expressed in terms of spherical Bessel functions j l (qr ) [12] (also see (10.15)) Rq,l (r ) = i

l

2m e q j l (qr ). π 2

(10.17)

The angular momentum l of the photoelectron wave is determined by the dipole selection rule that connects the core function of angular momentum c with the Bessel function of angular momentum l = c ± 1. The product q r is dimensionless and so is the Bessel function. The absolute value squared of the radial function, |Rq,l (r )|2 , has dimension [1/(energy×length3 )]. The spherical Bessel functions j l (qr ) that characterize the final states in K and L edge dipole excitations correspond to the lowest angular momenta l = 0, l = 1 and l = 2, respectively, and are given by3 2

A plane wave can be expressed as an infinite sum of spherical waves by the well-known expansion, eiq·r = 4π

l ∞

∗ il j l (qr ) Yl,m (θq , φq ) Yl,m (θr , φr ),

(10.15)

l=0 m=−l

which depends on the spherical angles of the electron wave vector q and the position vector r. 3 The spherical Bessel functions obey the closure relation ∞ j L (ar ) j L (br ) r 2 dr = 0

1 δ(a − b) 2a 2

(10.18)

494

10 Quantum Theory of X-Ray Absorption Spectroscopy

sin qr qr sin qr cos qr j1 (qr ) = 2 2 − q r qr

3 1 3 sin qr − 2 2 cos qr. j2 (qr ) = 3 3 − q r qr q r

j0 (qr ) =

(10.21)

10.3.2 Continuum Cross Section The x-ray absorption cross section is determined by dipole transitions from a core spin orbital to continuum states. The atomic core states have the form (10.10), core = n, c ,m c ,m s , of angular momentum c , whose radial function Rn c (r ) is given by (10.12). The continuum state q,l,m,m s is given by (10.16) and its radial part Rq,l (r ) by (10.17). The dipole operator does not act on spin so that m s = m s and we can drop the spin label and multiply the transition rate by 2 to account for the two spin states. The orbital momenta are linked by the dipole selection rule l = c ± 1. The polarization ( p = 0, +, −) dependent x-ray absorption cross-section (10.6) becomes p

σXAS = 8π 2 ω αf

c

   q,l,m | r ·  p |n, ,m 2 ρ(Ekin ) δ[ω − (Ekin + Ec )]. c c

m c =− c

(10.22) The energy balance reflected by the δ-function now contains the electron kinetic energy Ekin and core level binding energy Ec (taken positive), both referenced to the vacuum level. Their sum is equal to the photon energy ω. The density of final states per unit energy ρ(Ekin ) in (10.22) is taken care of by the normalization of the final state wavefunction. It yields ∞ ∞ ρ(Ekin ) δ(Ekin −E) dE = 2 q,l,m |q,l,m  δ(Ekin −E) dE = 2 −∞

(10.23)

0

orthogonality relation, ∞ j L (ar ) j L  (ar ) dr = 0

π δ(L , L  ) 2a(2L + 1)

(10.19)

and satisfy the asymptotic condition for r → ∞ j l (qr ) −→ .

  π(l + 1) 1 cos qr − . qr 2

(10.20)

10.3 Non-resonant Absorption: Excitation into Continuum States

495 (1)

Table 10.1 Angular momentum dipole matrix elements L , M| C p |l, m  (1) (l+1)2 −m 2 l + 1, m| C0 |l, m = (2l+3)(2l+1)  (1) l 2 −m 2  l − 1, m| C0 |l, m = (2l−1)(2l+1)  (1) (l+m+2)(l+m+1)  l + 1, m + 1| C1 |l, m = 2(2l+3)(2l+1)  (1) (l−m)(l−m−1)  l − 1, m + 1| C1 |l, m = − 2(2l−1)(2l+1)  (1) (l−m+2)(l−m+1)  l + 1, m − 1| C−1 |l, m = 2(2l+3)(2l+1)  (1) (l+m)(l+m−1)  l − 1, m − 1| C−1 |l, m = − 2(2l−1)(2l+1) ∗ Non-listed matrix elements are zero. The matrix elements are real, so that L , M|C (1) p |l, m = (1) (1) p L , M|C p |l, m = (−1) l, m|C− p |L , M. (1)

The matrix elements L , M| C p |l, m are identical to the coefficients ck=1 (L M; lm) tabulated by Condon and Shortley√[13] and Slater [14] and include the reduced matrix element l + 1||C (1) ||l = −l||C (1) ||l + 1 = l + 1 [11] listed in Appendix A.8.

which properly reflects the two possible spin states per electron. The polarization dependent matrix element is evaluated by writing the dipole operator as (10.24) r ·  p = r C (1) p . Here C (l) p with p = 0, +, − are the first order l = 1 (dipolar) subset of Racah’s spherical tensors Cm(l) [11], which are normalized versions of the spherical harmonics Yl,m = |l, m and are given up to l = 4 in Table A.2 in Appendix A.5. This notation facilitates evaluation of the polarization dependent transition matrix elements, L , M| C (1) p |l, m, which are listed in Table 10.1. The radial part of the matrix element R = b|r |a may be factored. When averaging over polarizations, the total matrix element is obtained as 2 1   q,l,m | r ·  p |n, c ,m c  3 m ,p c meq 2 = | j l (qr )| r |Rn c (r )|2 |l, m| C (1) p | c , m c | 3π2 m , p c   meq 2 2 = c | j c −1 (qr )| r |Rn c (r )| + ( c + 1)| j c +1 (qr )| r |Rn c (r )| . 3π2 (10.25) The angular momentum prefactors c and c +1 are the reduced angular matrix elements of the allowed c → c −1 and c → c +1 dipole transitions. Denoting the radial matrix elements as

496

10 Quantum Theory of X-Ray Absorption Spectroscopy

R2 c ±1 = | j c ±1 (qr )| r |Rn c (r )|2

(10.26)

and inserting into (10.22) we obtain4 σXAS =

 2m e q  8π ω αf 2 c R2 c −1 + ( c + 1)R2 c +1 . 3 

(10.28)

The dimensions are better seen by rewriting this expression in the form σXAS =

 q3  8π c R2 c −1 + ( c + 1)R2 c +1 . ω αf 3 Ekin

(10.29)

We see that in order for σXAS to have the proper dimension [length2 ], the squared matrix element R2 has the dimension [length5 ] (see footnote 4).

10.3.3 Simple Model Calculation The non-resonant XAS cross section from a specific core shell into continuum states given by (10.29) is also the photoemission spectroscopy (PES) cross section. It needs to be evaluated by use of good atomic wavefunctions. An approximate analytical solution, however, can be obtained by modified hydrogenic wavefunctions like those given by (10.12), where Z is taken to have an effective value Z ∗ < Z to account for screening effects. To illustrate the results obtained in such an approximation, we consider the K-shell cross section of Ne and L-shell cross section of Cu. For the K-shell, we have c = 0 and only the channel c → c + 1 is allowed. By use of hydrogenic core and plane wave final state wavefunctions we obtain σXAS = σPES =

256 (Z ∗ )5 a07 q 5 8π αf ω 1   . 3 Ekin (a0 q)2 + (Z ∗ )2 6   

(10.30)

2 R

Here a0 is the Bohr radius given by (10.13) and q the electron momentum, which is related by (10.14) to the kinetic energy of the photoelectron relative to the vacuum level. The first term has the dimension energy, and the middle term [1/energy] and 2 , distinguished by a tilde superscript, has the the redefined radial matrix element R 4

Our expression in SI units is the same as that derived by Cooper [4, 5] in atomic units given by σXAS =

  8π ω αf a02 c R 2c −1 + ( c + 1)R 2c +1 . 3

(10.27)

Here a0 is the Bohr radius, and the squared radial matrix elements denoted R 2 (instead of R in (10.28) ) have the dimension [1/energy].

10.3 Non-resonant Absorption: Excitation into Continuum States

1

Absorption cross section (Mb)

Fig. 10.1 K-edge excitation cross section for Ne, calculated with (10.30). By use of hydrogenic wavefunctions, the actual number of electrons Z = 10 in Ne needs to be reduced to Z ∗ = 6.5 to account for screening effects in order to better approximate the energy dependence

497

Ne K edge

10

-1

Henke Z =10 Z*=6.5

-2

10

-3

10

1000

2000 3000 4000 Photon energy (eV)

5000

usual dimension [length2 ]. The K-shell cross section for Ne calculated with (10.30) for Z ∗ = Z = 10 (blue curve) and Z ∗ = 6.5 (red curve) is shown in Fig. 10.1 in comparison with the one obtained from the Henke-Gullikson tables (black curve). For the L-shell, 2 p ( c = 1) core elections may be dipole excited into either εs (l = 0) or εd (l = 2) continuum states. The cross section is obtained as σXAS =

    ∗ 2 12 (Z ∗ )7 a05 q 3 8π αf ω 1 2 2 2 4 4 ) −4(a q) + 512 a q . 2048 (Z   0 3 Ekin 4(a0 q)2 +(Z ∗ )2 8      0  2 p→εd 2 p→εs    2 R

(10.31) The cross Sect. (10.31) is plotted in Fig. 10.2, decomposed into the two dipole allowed channels. We show the results obtained for the true Z = Z ∗ = 29 value (blue) and also for Z ∗ = 15 (red) which most closely approximates the empirical cross section shown in black, obtained from the Henke-Gullikson compilation [15, 16]. Although, the ambiguity of the Z ∗ value limits the reliability of such calculations, they still provide insight into the relative strength of the two dipole channels, which shows the dominance of the 2 p → εd channel in agreement with more sophisticated calculations [17].

10.3.4 The Core Level Photoemission Spectrum and Its Linewidth The core level photoemission spectrum consists of a measurement of the number of emitted electrons as a function of their kinetic energy. The measured peak is

10 Quantum Theory of X-Ray Absorption Spectroscopy

Fig. 10.2 Cross section for L-edge excitation in Cu metal, calculated with (10.31). The two allowed dipole excitation channels 2 p → εs and 2 p → εd are shown separated for the two indicated choices of effective Z ∗ values

Absorption cross section (Mb)

498

Cu L edge

1

2p 10

-1

2p

d

2p

d

2p

s

s

-2

10

1000

Henke Z*=Z=29 Z*=15

2000

3000 4000 Photon energy (eV)

5000

0 centered at Ekin , determined by the difference of incident photon energy ω and the 0 = ω − I P. The distribution of the core electron ionization potential (IP), i.e. Ekin 0 number of electrons around Ekin defines the intrinsic PES linewidth (FWHM) Ekin . If the instrumental (Gaussian) resolution is negligibly small, the intrinsic Lorentzian PES linewidth has the value Ekin = , where  is related to the lifetime of the core hole τ according to the uncertainty relation  = /τ . We will see that this fundamental atom-specific relation also plays a key role in resonant x-ray absorption (see Sect. 10.4.1) and in x-ray emission (see Sect. 12.2.7).

The intrinsic Lorentzian core level photoemission linewidth has the (FWHM) value , where , listed in Table 10.2 for selected core shells, is related to the lifetime τ of the core hole by the uncertainty relation  = /τ . The core level PES spectrum is formally defined as the number of photoemitted electrons NPES , per solid angle, per time, and per kinetic energy according to d3 NPES 2 (/2)2 = 0 σXAS . 0 2 d dt dEkin π  (Ekin − Ekin ) + (/2)2   

(10.32)

ρ(Ekin )

Here 0 is the incident photon flux, σabs the continuum absorption (= photoemis0 = ω − I P the central kinetic energy of the photoemission sion) cross section, Ekin peak, and  (dimension [energy]) the FWHM of a Lorentzian distribution function determined by the core hole lifetime. The distribution function ρ(Ekin ) is normalized to give unity when integrated over kinetic energy

10.4 Resonant X-Ray Absorption

499

∞ ρ(Ekin ) dEkin = 1.

(10.33)

−∞

In the following Section we will consider the excitation of core electrons into a well-defined resonant state, consisting of an atomic Rydberg orbital or an unfilled molecular orbital. In this case, we have a resonant x-ray absorption process where the excited electron remains trapped and is not photo-emitted. We will see that, remarkably, both the core level PES spectrum and the resonant XAS spectrum have the same minimum widths given by the total decay width  associated with the core hole.

10.4 Resonant X-Ray Absorption The resonant x-ray absorption case is illustrated in Fig. 10.3 for the case of transitions between two well-defined electronic states. For this case, the absorption rate (10.6) expresses the transition rate between two sharp non-degenerate levels, with energy conservation expressed by the Dirac δ-function, defined through its sifting property ∞ ρ(Eb ) δ(ω − (Eb − Ea )) dEb = 1.

(10.34)

−∞

The final state in XAS is an excited state with a core hole and our formulation so far has neglected that the core hole has a finite lifetime. In quantum mechanics, the lifetime τ of the core hole state gives rise to an uncertainty of the excitation energy, given by the uncertainty relation  = /τ . This causes a finite energy width in the measurement of any absorption line.

10.4.1 Natural Linewidth of XAS Resonances In the x-ray regime, the core hole may be spontaneously filled by two types of decay processes during which the energy stored in the excited electronic state is converted either into emitted x-rays or Auger electrons. Since the final electronic configurations reached by x-ray and Auger decays differ, the two emission channels do not interfere and the spontaneous decay rate for x-ray emission  X / and for Auger emission  A / simply add to the total decay rate /, as illustrated in Fig. 10.3. We therefore obtain for the total decay energy width   = X + A .

(10.35)

500

10 Quantum Theory of X-Ray Absorption Spectroscopy

=

X

+

A

b A

X

FWHM

XAS

0

h

h

e-

a Fig. 10.3 Schematic of excitation and de-excitation processes in a two-level system described by an electronic energy separation of E0 = Eb − Ea . The probability of x-ray absorption depends on the so-called detuning energy |ω − E0 |. The absorption rate XAS / reflects the total “up” rate. Once the electron is in the excited state |b it can decay either by x-ray emission with a rate  X / or Auger decay with a rate  A /. The total decay energy width is given by  =  X +  A and, as indicated, corresponds to the FWHM of a Lorenzian

Through this equation, we have introduced the so-called natural linewidth  in an ad hoc fashion, as typically done in the literature. We shall come back to the meaning of (10.35) later and again in conjunction with the linewidth in x-ray emission in Sect. 12.2.5. Since excited electronic states decay exponentially in time, the corresponding lineshapes in the energy domain are Lorentzians, and by convention  represents the Lorentzian FWHM. The widths reflect the total decay rates into a given core hole and constitute a polarization average. In Table 10.2 we have listed semi-empirical values of  for the K-shell of low Z atoms and the L3 shell of the 3d transition metals. The listed  values are taken from the tabulation of Krause [19]. Other tabulations [20, 21] give values that may differ by about 20%. This uncertainty reflects possible errors introduced by unfolding the intrinsic Lorentzian from instrumental Gaussian lineshape contributions, comparison of linewidths measured by different experimental techniques (x-ray emission, photoemission, Auger emission, and x-ray absorption), as well as differences in atomic linewidths due to chemical bonding or oxidation states [22]. Also listed are the corresponding core shell binding energies [18], which correspond to the absorption thresholds of the K and L3 shells. They approximately reflect the core shell resonance energies E0  E B . We have plotted the dependence of the natural energy widths  as a function of the binding energies listed in Table 10.2 in Fig. 10.4 for the 1s (K-edge) and 2 p3/2 (L3 edge) core shells [18]. For the limited energy range of the two atomic series there is an approximately linear relation. More generally, for the K-shell of all elements in the periodic table, the scaling of  with binding energy E0 is approximately  ∝ E02 Z 4 [23].

10.4 Resonant X-Ray Absorption

501

Table 10.2 Core shell resonance or binding energies E0  E B [18] and natural Lorentzian decay linewidths (FWHM) , for the K-shell of low Z atoms and L3 shell of the 3d transition metals [19, 20] K-shell L3 -shell Z Element E0 [eV]  [meV] Z Element E0 [eV]  [meV] 6 7 8 9 10 11 12 13 14 15 16 17 18

C N O F Ne Na Mg Al Si P S Cl Ar

284 410 543 697 870 1071 1303 1559 1839 2149 2472 2833 3206

100 130 160 200 240 300 360 420 480 530 590 640 680

20 21 22 23 24 25 26 27 28 29 30

Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn

346 399 454 512 574 639 707 778 853 933 1022

170 190 220 240 270 320 360 430 480 560 650

Figure 10.4 shows a plot of  versus E0 0.8

(eV)

0.7

Total decay width

Fig. 10.4 Plot of the tabulated values for  versus resonance (binding) energies E0 for the K- and L3 -shells in Table 10.2

L3 shell

/

0.6

-4 0 ~7x10

0.5 0.4 K-shell

0.3

/ 0.2

0

~ 2x10 - 4

0.1 0

0

500

1000

1500

2000

Binding energy

2500

3000

3500

(eV)

10.4.2 The Natural Shape of XAS Resonances We may utilize the integration property (10.34) of the singular δ-function and replace it by a finite width Lorentzian lineshape function with the same integrated property. In principle, each empty spin orbital will give a unit integrated contribution. For a single final state we can then express the final state energy density as

502

10 Quantum Theory of X-Ray Absorption Spectroscopy

ρ(Eb ) =

2 (/2)2 . π  (ω − (Eb − Ea ))2 + (/2)2

(10.36)

The normalized Lorentzian function has the dimension [1/energy], a FWHM energy width , given by (10.35), and a resonance peak height of 2/π. It has unit area when integrated over energy according to ∞ ρ(Eb ) δ(ω − [Eb − Ea ]) d(ω) = 1.

(10.37)

−∞

The so-defined energy density ρ(Eb ) is per spin and accounts for each spin dependent transition to substates within the total manifold of final states denoted by |b in the absorption cross Sect. (10.6). The case where transitions to many different final states occur is therefore implicity included in the matrix element. If the matrix elements to all final states are the same, one may also simply multiply the expression for ρ(Eb ) by the number of empty states, as will be discussed in conjunction with the calculation of the matrix elements. By denoting the resonance energy as E0 = Eb − Ea and use of the dipole approximation, we can write the absorption rate (10.3) or absorption cross Sect. (10.6) in the convenient compact form p

abs =  pk Wa→b

λ2 ab (/2)2 . π  (ω−E0 )2 +(/2)2   

(10.38)

p

σXAS

Here we have identified the polarization dependent XAS cross section given for a transition from an electronic state |a to a state |b by p

p

σXAS =

λ2 ab (/2)2 . π  (ω−E0 )2 +(/2)2

(10.39) p

The polarization dependent dipole transition energy width ab (dimension [energy]) is given by p

ab =

8π 2 ω 2  λ  

|b|r ·  p |a|2 . αf     coupling atomicpart photon includes spin part

(10.40)

The form of the dipolar matrix element nicely exhibits the photon and electronic contributions coupled by the fine structure constant.

10.4 Resonant X-Ray Absorption

503

According to (10.38) the XAS rate is determined by the ratio of two transition p p rates, ab / and /. The rate ab / is the dipole transition rate that follows from first order perturbation theory, while according to (10.35), the rate / is the total p decay rate. We can identify  X in (10.35) as the dipolar rate ab . This leads to a remarkable result. Although only the photon channel is detected in an x-ray absorption experiment, the minimum measured XAS linewidth has a “ghost” contribution from the Auger channel. The origin of this astounding fact will be discussed in more detail in Sect. 12.2.5. p

Our formulation of x-ray absorption in terms of the two rates ab / and / is similar to that used for the description of Mössbauer resonance scattering by Hannon and Trammell [24]. This reveals the similarity between resonant electronic x-ray and nuclear γ -ray processes. In both cases there is a non-radiative linewidth contribution, with the Auger channel for x-rays replaced by the conversion electron channel for γ -rays, also emphasized by Röhlsberger [25]. Our formulation also directly reduces to the optical Weisskopf-Wigner result [26] for the total decay linewidth , which we will show now.

10.4.3 Natural Linewidth of Optical Versus X-Ray Transitions In the optical region, dissipative losses due to ionization followed by Auger-like decays may be absent. In this case “absorption” corresponds to out-of-beam scattering.5 One can then derive a quantum mechanical expression for  from (10.40). By inserting the expression for αf , specifying the incident polarization along x, and accounting for the spin dependence of the matrix element by a factor of 2, one obtains the Weisskopf-Wigner result for spontaneous emission [26, 28]  = ab =

1 4ω3 16π 2 |b|x|a|2 2 |b|e x|a| = ω . α f 4π 0 3c3    3 λ2

(10.41)

without spin

It is apparent that for ab =  the absorption rate and cross section given by p (10.38) are maximized and σXAS becomes a special case of the resonant atomic Breit–Wigner cross section (6.52) [12, 29] discussed in Sect. 6.5.1.

5

This is how Dirac pictured optical “absorption”, as discussed in Chapt. VII Sect. 53 in his book [27].

504

10 Quantum Theory of X-Ray Absorption Spectroscopy

10.4.4 The Dipole Matrix Element The polarization dependent transition matrix elements in (10.40) may be written in terms of Racah’s spherical tensor operators with the angular wavefunctions written in the form of normalized spherical harmonics according to [11] Cm( )

=

 ( ) ∗ ( ) Cm = (−1)m C−m .

4π Y ,m (θ, φ) , 2 + 1

(10.42)

Because of their importance in modern angular momentum theory, we have listed these operators for = 0, 1, 2, 3, 4 in the Appendix in Table A.2. The orthogonality of the spherical harmonics is expressed by 

 Yl,m |Yl  ,m  =  l, m |l  , m   = δl,l  δm,m  ,

(10.43)

and the spherical tensors leads to the normalization sum rules

  Y ,m 2 = 2 + 1 , 4π m=−



 ( ) 2 C  = 1. m

(10.44)

m=−

The dipole matrix elements may then be factored into a radial and angular part given by = 1 and m = p = 0, ±1 assuming the role of the polarization label, according to b|r ·  p |a = R b|ˆr ·  p |a = R b|C (1) p |a .

(10.45)

Here rˆ is a unit vector, and R = b|r |a is the radial transition matrix element linking two electronic shells with radial wavefunctions Rn,c and Rn  ,L defined according to (10.10) and (10.11), given by, ∞ R = b|r |a =

Rn∗ ,L r Rn,c r 2 dr .

(10.46)

0

R has the dimension [length], and we shall assume it to be constant over the resonance width. The radial matrix element is typically calculated by means of atomic Hartree Fock self-consistent field approaches [9] where the final state configuration contains the core hole. Solid-state effects can be ignored for the core states and may be taken into account for the valence states by renormalizing them within their appropriate Wigner-Seitz spheres [30]. We will discuss the empirically derived radial matrix elements for the L-shell 2 p → 3d excitation of the 3d transition metals in Sect. 10.5.3 (see Fig. 10.15) and for both the K-shell 1 p → 2 p and L-shell in Sect.12.4.1 where we combine x-ray absorption and emission data (see Fig. 12.11).

10.4 Resonant X-Ray Absorption

505

The spherical tensor notation in (10.45) facilitates evaluation of the polarization dependent transition matrix elements when the angular parts of the wavefunctions are written in spherical harmonics Y ,m = | , m [31]. The non-zero angular transition matrix elements in the dipole approximation, L , M| C (1) p |l, m, between the spherical harmonics states denoted Yl,m = |l, m are listed in Table 10.1.

10.4.5 One-Electron/Hole Model In the simplest intuitive picture of resonant x-ray absorption, a photon transfers its energy to a core electron and the electron is excited into an unoccupied electronic state. In this so-called one-electron model one simply follows what happens to the photoelectron. The matrix elements (10.45) then connect filled electronic core states of the form |a = |n, l, m, sz  with empty valence states |b = |n  , L , M, sz . The model ignores all other “passive” electrons in the atom during the excitation process. The only coupling of angular momenta of electrons that is included in this scheme is the spin-orbit coupling between the spin and orbital angular momenta of the single electron. Couplings of angular momenta between different electrons are neglected. The essence of this one-electron picture is illustrated in Fig. 10.5 a for the L-edge 2 p → 3d x-ray absorption spectrum of a 3d transition metal atom.

(b)

(a) One-electron picture (2p

1

1

3d electron transition)

6

5

10

Excitation

3d5/2 3d3/2

Electron

Hole energy

Electron energy

Excitation

2p3/2 2p1/2

Configuration picture 9

(2p 3d 2p 3d electron transition 1 1 or 3d 2p hole transition)

2p1/2 2p3/2

3d3/2 3d5/2

Hole

Fig. 10.5 Description of resonant L-edge x-ray absorption in two pictures, a one-electron picture and b d 9 configuration picture. Each energy level is labeled by its quantum numbers n, l, j = l ± s. In a the quantum numbers label one-electron states, in b they label hole states for the special case of a 3d 9 electronic ground state, containing nine electrons or equivalently one hole. The two pictures are equivalent for the description of the electronic transitions, but the sign of the angular part of the dipole matrix elements may be different, as discussed in the text

506

10 Quantum Theory of X-Ray Absorption Spectroscopy

Table 10.3 Spin-orbit wave functions for p states One-elec. label | j, m j  basis j j mj p1

1 2

p3

3 2

2

+ 21 − 21

2

+ 23 + 21

| , m, s, m s  basis Y ,m ↑ or ↓   √ √1 −Y1,0↑ + 2 Y1,+1 ↓ 3   √ √1 − 2 Y1,−1↑ +Y1,0↓ 3

Y1,+1↑ √ 1 √

 2 Y1,0↑ +Y1,+1↓   √ Y1,−1↑ + 2 Y1,0↓

− 21

3 √1 3

− 23

Y1,−1↓

The electron spin wavefunctions are |s, 1/2 = ↑ (spin-up) and |s, −1/2 = ↓ (spin-down)

In this model the “initial state” is envisioned as an electron in the 2 p core state. The active electron has an angular momentum l = 1 and spin s = 1/2, so that the spin-orbit coupling given by (9.7) produces two energy states with j = l ± s. The states 2 p3/2 with j = 3/2 and 2 p1/2 with j = 1/2 are separated by E = 3ξ2 p /2, with ξ2 p given in Table 9.1. The core spin-orbit splitting gives rise to the prominent L3 (2 p3/2 → 3d) and L2 (2 p1/2 → 3d) edges in the experimental spectra in Fig. 7.23. In the “final state” the active electron is located in the 3d shell with angular momentum l = 2 and spin s = 1/2. The spin-orbit coupling again produces two energy states with j = l ± s, i.e. the states 3d5/2 with j = 5/2 and 3d3/2 with j = 3/2, as shown in Fig. 10.5a. Because the valence shell is less compact, its spin-orbit coupling constant ξ3d is about 100 times smaller than ξ2 p as seen from Table 9.1. For the calculation of the transition matrix element, one expresses the angular part of the spin-orbit spit core states as a linear combination of spherical harmonics. As an example, we have listed in Table 10.3 the m j spin-orbit substates of the p states.6 The angular part of the transition matrix elements between the spin-orbit core states and the valence states can then be calculated from matrix elements involving core and valence states expressed in terms of spherical harmonic basis functions under the assumption of spin conservation in the transition since the dipole operator does not act on spin. The coefficients a and b in the functions | jm j  = a|l, m ↑ + b|l, m + 1 ↓ listed in Table 10.3 are the real Clebsch–Gordon or 3 j Wigner coefficients which enter as weight factors in the transition matrix elements, as discussed later. The one-electron picture of Fig. 10.5a is misleading in that it depicts the spin-orbit splitting of the p core shell as an “initial state” effect. In reality the p shell is filled in the ground state and there is no spin-orbit splitting for a closed shell. In the proper description of the x-ray absorption process, based on a configuration picture shown in Fig. 10.5b, an atom is excited from a ground or initial state configuration to an excited or final state configuration. In discussing transitions between configurations one typically omits all closed subshells since they are spherically symmetrical and their net angular momentum is zero [9]. Listing only the active shells, resonant L6

In Appendix A.7 we give the spin-orbit functions of all s, p, and d states in Table A.5.

10.4 Resonant X-Ray Absorption

507

edge absorption is described by an initial state electron configuration 2 p 6 3d n and a final state configuration 2 p 5 3d n+1 . The problem with the one-electron description now emerges. Instead of properly including the couplings of all angular momenta within the open 3d n valence shell in the ground state and the couplings of electrons in the open 2 p and 3d shells in the final state configuration 2 p 5 3d n+1 , the one-electron picture ignores all these couplings. Instead one considers the simplest case where the initial state contains a filled p 6 shell and nine electrons in the d shell. Now one pictures the d 9 electron configuration in a hole picture, which is simply a d 1 hole configuration. In the configuration model, the initial state is then the d 1 hole configuration. The final state p 5 d 10 has a filled d-shell, and a p 5 electron is equivalent to a p 1 hole configuration. Hence for a d 9 ground state, L-edge spectra in the configuration picture are described by a transition between the hole configurations d 1 → p 1 . The one-electron model has turned into an inverted one-hole model as shown in Fig. 10.5b. Now the spin-orbit splitting of the p shell and the presence of a core hole are properly described as final state effects. For the description of valence states that contain more than single electrons or holes one may simply account for the number of electrons or holes by a filling factor. For example, a d 2 valence electron configuration with eight holes will have a four times larger resonant cross section than a d 8 electron configuration with two holes. This is in remarkable agreement with experimental L edge spectra of the elementary 3d transition metals as shown in Fig. 10.16, below. The power of the oneelectron/hole model lies in the description of solids with weak electron correlations. In such cases, one may simply describe the valence properties in an independent electron band-like model as discussed in Sects. 10.5.3 and 11.5.1.1. The one-electron model or its inverted one-hole scheme only considers the intrinsic coupling of the spin and angular momentum of independent electrons and ignores multi-electron coupling effects. Its wide spread use is based on its extension to the description of x-ray absorption in weakly correlated solids, with the valence electrons treated in an independent electron model by band or density functional theory. The different signs of the charge of electrons and holes lead to some differences that are easily accounted for. For example, the energy order of the j states is inverted in the two schemes in Fig. 10.5 because electrons and holes have opposite spin.7 Also the polarization dependent angular matrix elements may have opposite signs, which however, disappears when the transition probability is calculated as the absolute value squared.

7 This is expressed by Hund’s third rule stating that the electronic ground state has the minimum possible j value for a less than half filled shell and the maximum possible j value for a more than half full shell.

508

10 Quantum Theory of X-Ray Absorption Spectroscopy

10.4.6 Polarization Dependence of the Angular Transition Matrix Element The angular transition matrix elements may similarly be given in a one-electron picture or the upside-down hole picture. The only difference is that certain matrix elements change sign because of the inverted motion of electrons and holes in Fig. 10.5. This is readily seen from the form of the matrix elements in Table 10.1 which depend on the direction of the transition according L , M|C (1) p |l, m = (−1) p l, m|C−(1)p |L , M for p = +1, 0, −1. For convenience we give the values for the non-zero angular matrix elements between s, p, and d states in Fig. 10.6 (also see Fig. A.8). The tensor matrix elements listed in Fig. 10.6 obey sum rules which follow from the orthogonality of the spherical harmonics and the quantum mechanical rule that only indistinguishable paths between initial and final states can interfere. Since all transitions in the figure involve individual lower and upper states, the listed amplitudes do not interfere. The individual and summed transition probabilities are therefore just the square of the listed amplitudes. The listed transition matrix elements do not include the spin degeneracy which is added by consideration of strict spin conservation in dipole transitions. For a non-magnetic sample or a magnetic sample

Y1,n | C i | Y0, 0 (1)

x

Y1,n | C i | Y2, (1)

3

x

45

Y1, 1

Y1, 1 Y1, 0 p Y1,-1

p Y1, 0

Y1,-1

i= +

0

-

+

1

1

1

- 3 - 9 - 18

s Y0, 0

0

-

9 12 9 - 18 - 9 - 3

Y2, 2 Y2, 1 Y2, 0 Y2,-1 Y2,-2

d

Fig. 10.6 Angular transition matrix elements between s, p, and d states, expressed in terms of their spherical harmonics components Yl,m = |l, m, using spherical tensor transition operators (1) Ci . The matrix elements are calculated by use of Table 10.1, and they depend on the direction of (1) (1) the transition according to L , M|Ci |l, m = (−1)i l, m|C−i |L , M for i = +1, 0, −1

10.4 Resonant X-Ray Absorption

509

with random domains one may therefore calculate the core to valence transition probability based on orbital degeneracy, only, and then account for the spin degeneracy by multiplying the final result by 2.

10.4.7 Sum Rules for the Angular Transition Matrix Element The transition probabilities increase with the total number of ways that transitions may occur. In practice, this is determined by the degeneracies and occupancy of the 2c + 1 core and 2L + 1 valence orbitals plus the three possible polarization orientations. In a Cartesian coordinate system, the polarization dependent matrix elements given in Fig. 10.6 correspond to the values p = ±1 for circularly polarized photons with positive and negative helicities propagating into the z direction (k z), and p = 0 describes linearly polarized photons with polarization vector E z as illustrated in Fig. 10.7. We first state a general theorem for the absolute value squared of various types of sums of dipole matrix elements 2      (1) L , M|C (1) |c, m2 .  L , M|Cq |c, m = q 

(10.47)

There are no cross terms, and we can treat all individual angular momentum matrix elements as orthogonal. The polarization dependent transition probabilities

p= h = 0

p= h =+1

E,z

k,z

E

k

L

p= h =-1

k,z

E L

z Fig. 10.7 Three orthogonal polarized photon states corresponding to the polarization labels p = 0, +1, −1 which also reflect the photon angular momenta 0 + , − according to Fig. 3.4a. In a Cartesian (x, y, z) coordinate system the three cases correspond to linearly polarized photons with their E-vector along z and circularly polarized photons for k along z and the shown angular momentum directions L. Underneath we have identified the associated unit polarization vectors, defined in (3.30)

510

10 Quantum Theory of X-Ray Absorption Spectroscopy

Transition Intensity Sum Rules c 2L+1

2L+1

p

L 3

L

2c+1

2c+1

L=c+1 2L+1

any substate

p

p

L

L

2c +1

2L+1

any substate

2c+1

Fig. 10.8 Sum rules for the square of the angular transition matrix element for transitions between core states with angular momentum c, m and a total number of 2c + 1 core orbitals and valence states with angular momentum L , M, and 2L + 1 orbitals, assuming L = c + 1. The x-ray polarization is represented by a label p = 0, ±1. All sums over polarization and manifolds are indicated graphically by a loop. When a sum is performed over at least two of the three labels M, m, p, the transition matrix element is independent of polarization and yields the sum rule value given in red

(1) 2 2 |L , M|C (1) p |c, m| = |c, m|C − p |L , M| then fulfill the following sum rules (without spin)

2 |L , M|C (1) p |c, m| = L

(10.48)

2 |L , M|C (1) p |c, m| =

L 3

(10.49)

2 |L , M|C (1) p |c, m| =

L 2L + 1

(10.50)

2 |L , M|C (1) p |c, m| =

L . 2c + 1

(10.51)

m,M, p

m,M

m, p

M, p

By inclusion of spin we need to consider that the dipole operator does not act on spin, so that only spin-conserving transitions are allowed. To include spin, one therefore needs to first calculate the matrix elements according to the above orbital degeneracy sum rules and then multiply by a factor of 2. The sum rules are graphically illustrated in Fig. 10.8, where a loop indicates a sum over m, M, or p.

10.4 Resonant X-Ray Absorption

511

Table 10.4 Mathematical description of s, p and d orbitals in terms of spherical harmonics Yl,m l = |l, m √1 = s= Y0,0 4π 3 x √1 ( Y1,−1 − Y1,+1 ) px = = 2  4π r 3 y √i (Y1,−1 + Y1,+1 ) = py = 2  4π r 3 z = Y pz = 1,0  4π r 15 x y √i (Y2,−2 − Y2,+2 ) dx y = = 2  4π r 2 15 x z √1 (Y2,−1 − Y2,+1 ) dx z = = 2 2  4π r 15 yz √i (Y2,−1 + Y2,+1 ) = d yz = 2 2  4π r 2 2 15 (x −y ) √1 (Y2,−2 + Y2,+2 ) dx 2 −y 2 = = 2 16π r 2  5 (3z 2 −r 2 ) d3z 2 −r 2 = = Y 2,0 2 16π r   2π π The orbitals oi are real and normalized according to |oi |2 d = 0 0 |oi |2 sin θ dθ dφ = oi |oi  = 1

By use of (10.40), (10.45), and (10.49) we obtain by including spin p

c→L =

8π 2 ω αf 2 16π 2 ω αf 2 2 Rc→L 2 |L , M|C (1) Rc→L L .(10.52) p |c, m| = 2 λ 3λ2 m,M

This expression assumes that all 2(2c + 1) core spin-orbital substates are filled and all 2(2L + 1) valence substates are empty. As shown by the last expression, there is no polarization dependence in this case (sum rule (10.49)). This is a consequence of the fact that closed shells are spherically symmetric as expressed by the sum rules (10.44). Sum rules also exist when the complex spherical harmonics are combined into the real molecular orbitals listed in Table 10.4, which is typically done to describe bonding of atoms in molecules and solids. Transition matrix elements between the p3/2 and p1/2 core states and the di orbitals are shown in Fig. A.8. The symmetry properties of different sums of orbitals listed in Tables 10.3 and 10.4 are illustrated in Fig. 10.9, where we have plotted charge densities (absolute values squared of the wavefunctions) for selected cases. The top rows show that the charge densities of sums of the p orbitals and their spin-orbit split p1/2 and p3/2 submanifolds are all spherically symmetric. The spherical symmetry of the spin-orbit split submanifolds allows us to separately consider sum rules for the transitions from the 2 p1/2 (L2 ) and 2 p3/2 (L3 ) core states. Similar symmetry rules apply for the d orbitals, shown in the lower rows of Fig. 10.9, and also for their spin-orbit split d3/2 and d5/2 submanifolds (not shown). The spherical symmetry of the summed d orbitals is preserved even when they are split into the well-known eg and t2g substates by a cubic ligand field, as shown in the figure. As a consequence we may state the following rules for the polarization dependence of core to valence transitions.

512

10 Quantum Theory of X-Ray Absorption Spectroscopy

Fig. 10.9 Illustration of the charge densities of various wavefunctions in Tables 10.3 and 10.4 and their sums. On top we show that the charge densities of sums of the p and their spin-orbit split p1/2 and p3/2 submanifolds are all spherically symmetric. In the lower panel we show the five di orbital densities whose sum is again spherically symmetric. The spherical symmetry is preserved even when they are split into the eg and t2g irreducible representations by a cubic ligand field

p orbitals

z x

px

y

sum

pz

py

p1/2 m j = +1/2

-1/2

p3/2 +3/2

-3/2

-1/2

+1/2

d orbitals

eg d3z - r 2

dx

2

2

- y2

t2g dxy

dxz

d yz

There is no polarization dependence of core to valence transitions whenever the charge densities of the core states and valence states are spherically symmetric. This applies for all free atoms and molecules and for solids with cubic symmetry and random bond orientations due to polycrystallinity or lack of orientational order in soft matter. In the presence of magnetic order, the polarization dependence also vanishes for samples with random orientations of magnetic domains. More generally, the polarization average is defined as c→L =

 1 p 1 0 +1 −1 ,  = + c→L + c→L  3 p c→L 3 c→L

(10.53)

0 and if the transition matrix element is polarization independent, we have c→L = +1 −1 = c→L and therefore c→L 0 = c→L = c→L

 1  +1 −1 . c→L + c→L 2

(10.54)

10.5 Resonant XAS in Experiment and Theory

513

In cases where an orientational average of the sample exists, (10.52) may be generalized to the case where the valence states are partially filled. In this case we may define an effective number of empty or “hole” states given by the fraction Nh /(2(2L + 1)). The valence charge then remains effectively spherically symmetric, and we can rewrite (10.52) and the associated cross section in the following form that we will consider in the next section. For a sample where an orientational average can be made and only a fraction of the valence states Nh /(2(2L + 1)) is unfilled, the resonant XAS cross section and the transition matrix element are given by σXAS =

λ2 c→L (/2)2 , π  (ω − E0 )2 + (/2)2

(10.55)

where including spin we have c→L =

L Nh 8π 2 ω αf 2 Rc→L . λ2 3(2L + 1)

(10.56)

In the following we shall compare experimental data with this quantum theoretical expression for the x-ray absorption cross section.

10.5 Resonant XAS in Experiment and Theory In this section we present selected experimental K-shell absorption spectra for low-Z atoms and L-shell spectra for the 3d transition metals. Our choice of these cases is due to their practical importance, their large cross sections, and sharp absorption lines [31–34].

10.5.1 K-Shell Resonance in the Low-Z Atom Ne The K-shell absorption spectrum of Ne atoms consists of a series of Rydberg resonances that smoothly merge into a continuum around the ionization potential (IP) as illustrated in Fig. 10.10, which shows the experimentally determined cross section [35]. We first consider the linewidths of the Rydberg resonances. The experimental data shown as circular points are plotted as the absorption cross section on a logarithmic scale to better reveal the fitted Voigt lineshapes (see Appendix A.2.3). The cross section of each XAS Rydberg resonance is described by (10.55) with specific dipole transition matrix elements 1s→np and FWHM linewidths  = (np). The natural

10 Quantum Theory of X-Ray Absorption Spectroscopy

1

X-ray absorption Ne K edge Rydberg series

IP

Cross section

abs

(Mb)

514

0.1

0.01

864

865

866

867

868

869

870

871

Photon energy (eV)

Fig. 10.10 High resolution x-ray absorption spectrum of the K edge of Ne, revealing the series of Rydberg resonances that merge into the ionization potential (IP) of 870.2 eV [35]. The cross section is plotted on a logarithmic scale to emphasize the Voigt profiles of near Lorentzian shapes of the Rydberg resonances, shown as dotted green lines and the continuum step shown as a dashed blue line

linewidths (np) were determined by Müller et al. [35] from Voigt lineshape fits with an experimental Gaussian resolution function 32 meV as (3 p) = 248(2) meV , (4 p) = 260(3) meV and (5 p) = 297(5) meV, respectively. The 3 p value is in accord with the semi-empirical value of 240 meV in Table 10.2 and compares with the Ne 1s core level photoemission linewidth of 270 meV (see footnote 1 in [36]) and the Kα x-ray emission linewidth value of 270 meV [37] (see Fig. 12.6). Next we consider the value of the dipole matrix element 1s→3 p for the first Rydberg resonance in Fig. 10.10. The final state has 3 p character, and all six 3 p (L = 1) spin orbitals are empty so that Nh = 2(2L + 1) = 6. The peak value of the cross section (10.55) is obtained as res = σXAS

λ2 1s→3 p 16πω αf 2 = R1s→3 p . π  3

(10.57)

Using the resonant wavelength of ω = 867.3 eV (λ = 1.43 nm) and the experimenres = 2 Mb = 2 × 10−4 nm2 tal values  = 248 meV and the measured peak value σXAS we can deduce the value of the radial matrix element to be R21s→3 p =

3 σ res = 4.7 × 10−7 nm2 16πω αf XAS

(10.58)

or R1s→3 p = 6.8 × 10−4 nm. We will see in Sect. 12.4 (see Fig. 12.11) that this 1s → 3 p radial matrix element is smaller than the 1s → 2 p one by a factor of about 6. This is expected due to the smaller radial overlap of the 1s core state with the more extended 3 p state relative to the more compact 2 p state.

10.5 Resonant XAS in Experiment and Theory

515

The dipolar transition matrix element is evaluated to be 1s→3 p = 7.6 × 10−2 meV. The ratio of 1s→3 p /   3 × 10−4 reveals the well-known fact that the dominant linewidth contribution originates from the Auger decay channel.

10.5.2 K-Shell Resonances in the N2 and O2 Molecules Two other fundamental examples of resonant x-ray absorption spectra are the strong K-shell XAS resonances in the N2 and O2 molecules. We shall later in Sect. 13.10.1 also discuss their high-resolution resonant x-ray emission spectra. They are described by a resonant scattering process where the XAS final state is an intermediate state in the scattering process.

10.5.2.1

N2 Molecule

We start with the well-known XAS spectrum of N2 , whose fine structure was first observed by electron energy loss spectroscopy [38] in the late 1970s [39, 40]. It was reproduced with x-rays only about 10 years later [41]. The electronic orbital structure of N2 , its spin dependent electron filling, and the origin of the vibrational fine structure in the K-shell 2σu → 1πg∗ resonance are illustrated in Fig. 10.11 together with experimental data recorded by Prince et al. [42]. The N2 XAS spectrum is determined by transitions illustrated with increasing detail in Figs. 10.11a–c between the electronic ground state, whose lowest vibrational state ν = 0 is the only one occupied at room temperature and the shown vibrational states in the electronic final state [43]. The experimental XAS spectrum [42] is shown in Figs. 10.11d. A detailed lineshape fit yielded a core hole lifetime width of  = 115 ± 4 meV [21, 42], which compares to the semi-empirical value of 130 meV in Table 10.2. The origin of the relative intensities of the vibrational XAS resonances, separated by  230 meV, is illustrated in (c). There we show the vibrational wavefunctions of the electronic ground and final state in the internuclear distance dependent Morse potentials [45], taken from [43, 44]. The gray areas in (c) represent the integrals of the products of the vibrational wavefunctions of the ν = 0 ground state with those of the excited states ν = 0, 1, 2.... The square of the gray areas is the so-called Franck–Condon overlap factors [46] which determine the relative intensities of the vibrational XAS resonances.

10.5.2.2

O2 Molecule

For comparison, we show in Fig. 10.12 the case of O2 , again illustrating the spin dependent filling of the molecular orbitals, and the origin of the same K-shell 1σu →

516

10 Quantum Theory of X-Ray Absorption Spectroscopy

(a) N2 spin orbital (b) XAS configuration diagram diagram

1

3 *u

3 1

=3 =2 =1 =0

g

2

u

2

g

final state 401.4

=2

~120meV

u

h

+1

1 *g

final state

~230meV

1 *g

-1

u

Energy (eV)

2p

(c) Vibrational states in Morse potential

with vibrational splittings

h

401.2

=1 401.0 400.8

=0

2s ~290meV 0.1eV

=1

u g

=0

Cross section (Mb/atom)

1s 1 1

ground state

8 (d)

N2 1

u

1 *g

absorption resonance

6 4 2 230 meV

0 400

401 402 Incident photon energy (eV)

403

Fig. 10.11 a Molecular orbital diagram of the N2 molecule, with spin dependent filling in the ground state indicated by black (spin-up) and red (spin-down) circles. The dipole and parity allowed 1σu → 1πg∗ spin-conserving transition is also shown. b XAS transition diagram in a configuration model. With increasing photon energy transitions may be observed to all vibrational states in the final state with a hole in the 1σu and an added electron in the 1πg∗ orbital. The natural Lorentzian linewidth (FWHM) of the final states is   120 meV [21, 42]. c Vibrational wavefunctions of the ground state and the first three vibrational final states, with shaded areas indicating products of ground and final state functions [43, 44]. d High-resolution XAS spectrum of the 1σu → 1πg∗ resonance in N2 [42]. The intensity scale has been converted to the cross section per atom. The vibrational splitting of the resonances is  230 meV, and their relative intensities are given by the square of the gray areas in (c)

10.5 Resonant XAS in Experiment and Theory

517

1πg∗ resonance as in N2 , using experimental data of [21]. For a precise calibration of the energy of the 1σu → 1πg∗ and higher energy Rydberg transitions we refer the reader to the interesting article by Leutenegger et al. [47]. The electronic orbital structure of O2 , its spin dependent electron filling, and the origin of the K-shell resonance are illustrated in Figs. 10.12a, b. The 1s core state is split into the 1σu and 1σg molecular components with a splitting of 1.11 eV [48]. The molecular orbital structure of N2 and O2 is very similar, the main difference being the filling of the 1πg∗ oribital which is empty in N2 and in O2 contains two electrons of the same spin according to Hund’s rule. The molecule is hence paramagnetic. The dipole operator does not act on spin so that the two orbitals involved in a transition must have the same spin. We have arbitrarily assumed excitations of “spin-down” electrons in Figs. 10.12a. Since the dipole operator has odd parity, parity conservation demands that the lowest energy allowed transition is also parity odd [32]. The electronic transition of interest indicated by a red arrow connects the 1σu spin-down K-shell orbital and the empty 1πg∗ orbital of the same spin. The experimental spectrum is shown in Fig. 10.12c [21], plotted in terms of the x-ray absorption cross section per atom. The cross section was obtained by conversion of the oscillator strength of the inner shell electron energy loss spectra (ISEELS) [40] and (A. Hitchcock private communication). The spectrum reveals partially unresolved vibrational finestructure8 which is more clearly revealed by the shown fitted Voigt profiles consisting of Gaussian 55 ± 10 meV and Lorentzian 149 ± 10 meV FWHM components. The implied natural linewidth of the 1s core hole state of 1s = 149 ± 10 meV agrees rather well with the semi-empirical value of 160 meV in Table 10.2.

10.5.3 L-Shell Resonances in 3d Transition Metal Atoms Other important resonances involve the so-called white-lines9 in the L absorption spectra of the 3d transition metals. They were already discussed in Chapt. 7 and shown in Figs. 7.12 and 7.24, and their study has played an important role in modern magnetism studies of thin films and nanostructures [31]. We note that with the advent of low energy FELs and up-converted laser sources that cover a limited spectral range up to about 100 eV (see Fig. 1.4) with femtosecond time resolution, it has become popular to study the dynamical processes of 3d transition metals by M-shell (3 p → 3d) absorption spectroscopy in the 30–80 eV range [50, 51], instead of L-shell (2 p → 3d) spectroscopy in the 300–1000 eV, discussed here. The advantage of L-edge studies is the about eight times larger 2 p spin-orbit splitting which leads to a reduction of multi-electron correlation effects. The Lshell resonances also exhibit larger resonances of smaller widths, since M-shell 8 9

A higher resolution O2 XAS spectrum has been recorded by Strocov et al. [49]. In early studies the intense resonances appeared as white lines on photographic films.

518

10 Quantum Theory of X-Ray Absorption Spectroscopy

(a) O2 spin orbital diagram

(b) XAS configuration diagram with vibrational splittings

3 *u

1

-1

u

+1

1 *g

final state

2p

1 *g 1 3

~125meV =2 =1 =0

u g

h 2s

2 2

~150meV

h

u g

~200meV

Cross section (Mb/atom)

1.1eV

1s

1 1

6

(c)

u g

=1 =0

ground state

O2 1

5 125 meV

u

1 *g

absorption resonance

4 3 2 1 0 530.0 530.5 531.0 531.5 Incident photon energy (eV)

Fig. 10.12 a Molecular orbital diagram of the O2 molecule, with spin dependent orbital filling in the ground state indicated by black (spin-up) and red (spin-down) circles. Indicated in red is the 1σu → 1πg∗ transition of interest, where an electron from the K-shell 1σu orbital (arbitrarily assumed to be spin-down) is dipole exited to the lowest unfilled 1πg∗ orbital of the same spin. b XAS transition diagram in a configuration model. At room temperature only the lowest vibrational ground state is filled and with increasing photon energy transitions may be observed to all vibrational states in the final state with a hole in the spin down 1σu and an added electron in the 1πg∗ orbital. c High-resolution x-ray absorption spectrum of the 1σu → 1πg∗ resonance in the O2 molecule [21], with the intensity scale converted to the cross section per atom. The vibrational fine structure with splitting ∼ 125 meV has been fitted with the shown Voigt profiles containing instrumental Gaussian and Lorentzian lifetime contributions [21]. The O2 K-shell natural linewidth is obtained from the width of the vibrational lineshapes to be  = 150 meV

10.5 Resonant XAS in Experiment and Theory

519

XAS cross section (Mb)

0.04

0.03

Fe metal K-edge

0.03

Co metal K-edge

0.02 0.02 0.01

0.01

0

0 7100

7120

7140

7160

7700 7180 Photon energy (eV)

7720

7740

7760

7780

Fig. 10.13 K-shell XAS spectra of Fe and Co metal (data from Alain Fontaine private communication). We have subtracted a pre-edge background and put the data on an absolute cross section scale that only reflects the K-shell contribution by comparison with tabulated cross sections [15]

resonances are lifetime broadened due to the presence of super-Coster-Kronig decays, which occur between shells of the same principal quantum number [52, 53]. For example, the natural M3 linewidth for Co is  = 2.85 eV compared to the L3 width of  = 0.43 eV. The difference between the M and L spectra of the 3d transition metals is most beautifully demonstrated by the spectra of the elementary atoms and cations in [54–56]. L-shell XAS spectra of the transition metals also have an advantage over K-shell spectra (1s → 4 p transitions) in that their strong resonant structure is absent at the K-shell thresholds as illustrated for Fe and Co and in Fig. 10.13. Furthermore, the absolute K-shell cross section, also indicated in that figure, is about two orders of magnitude weaker than for the L-edges. This is more directly apparent from Fig. 7.14 which shows the Fe XAS cross section over an extended energy scale. For L-edges, electrons from the 2 p1/2 (L2 ) and 2 p3/2 (L3 ) core states are excited to empty 3d valence states. In a simple density of states model the valence shell can be described in terms of the number of filled “electron” orbitals, n e , and empty “hole” orbitals, n h , so that for each spin we have n e + n h = 2L + 1, as shown in Fig. 10.14. The model assumes that at a given energy relative to the Fermi level, the density of states contains equal contributions from all angular momentum substates −L ≤ M ≤ L due to hybridization (see Fig. 11.10). This is a reasonable approximation for metallic bonding even if the lattice deviates somewhat from cubic symmetry as in hexagonal close packed (hcp) Co metal. This mixing of different angular momentum substates M eliminates any polarization dependence. The spin dependence is also averaged out if the sample contains microscopic magnetic domains of random orientation. We consider the total resonant cross section for 2 p → 3d excitations, i.e. the added L2 and L3 contributions. The angular parts of the transition matrix elements follow from the sum rules (10.48)–(10.51). The spin-inclusive expression is simply

520

10 Quantum Theory of X-Ray Absorption Spectroscopy

Absorption c

L=c +1 # empty

valence state |b

nh orbitals

orb.degeneracy

ne # filled

nh+ne= 2L+1

orbitals

L nh 3 2L+1

core state |a orb.degeneracy 2c +1

all orbitals filled

Fig. 10.14 Probability of dipole transitions from a core manifold of angular momentum c to a valence manifold of angular momentum L = c + 1. The core manifold consists of 2c + 1 (−c ≤ m c ≤ c) orbitals and the valence manifold of 2L + 1 orbitals between which spin-conserving transitions can occur. For the valence orbitals we have assumed that a “hole” fraction n h /(2L + 1) of orbitals is empty and a fraction n e /(2L + 1) is filled with electrons. The transition probability is independent of polarization and given in red

a factor of 2 larger since there are two identical charge scenarios for each spin, and spin-flip transitions are dipole forbidden. In anticipation of the discussion of magnetic effects in Sect. 11.5 we can include spin in the total number of filled and empty states by replacing the number of orbitals by the number of spin orbitals according to Nh = 2n h and Ne = 2n e , so that Ne + Nh = 2(2L + 1) reflects the total substates of a shell of angular momentum L. The total polarization averaged dipole transition width is then obtained from (10.56) by considering the partial occupancy Nh /(2(2L + 1)) of the valence shell as c→L =

L Nh 8π 2 ω αf 2 Rc→L , 2 λ 3(2L + 1)

(10.59)

and the resonant XAS cross section (10.55) becomes σXAS =

L Nh 8π ω αf (/2)2 R2 .  (ω−E0 )2 +(/2)2 c→L 3(2L + 1)

(10.60)

We are also interested in the energy integrated value of the resonant cross section which has the units [Mb eV]. This number can be determined from experimental data. We approximate the energy integration of the resonant cross section by integrating over the Lorentzian lineshape at a constant central energy to obtain

10.5 Resonant XAS in Experiment and Theory



∞ 0

521

L σXAS d(ω) = 4π 2 ω αf R2c→L Nh . 3(2L + 1)   

(10.61)

Cc→L

This is the same expression given in [31]. It is important to realize that Nh reflects the number of empty states in the electronic ground state, i.e. in the absence of a core hole. This is a consequence of the fact that in the proper configuration model discussed in conjunction with Fig. 10.5 the core hole state is a final state effect. This is also expressed by the so-called initial and final states rules in x-ray absorption and emission spectroscopies [57, 58]. For the 2 p → 3d L-shell resonances of the transition metal atoms the radial transition matrix element is then given by R22 p→3d =

15 C2 p→3d . 8π 2 ω αf

(10.62)

1.0x10

-2

7.5x10 L-shell 2p 3d

0.9x10 -2 -2

0.7x10

-2

7.0x10-2 6.5x10

-2

6.0x10-2

0.8x10

-2

8.0x10 -2

M-shell 3p 3d

0.6x10-2

20 21 22 23 24 25 26 27 28 29 Atomic number Z and Element

5.5x10

-2

5.0x10

-2

4.5x10-2 4.0x10 -2

(nm)

-2

V Cr Mn Fe Co Ni Cu

3d

1.1x10

Ca Sc Ti

Radial matrix element R 3p

(nm)

-2

Radial matrix element R 2p

1.2x10

3d

We can use this expression and the conversion 1 nm2 = 104 Mb to determine the values of R2 p→3d for the 3d transition metal atoms by use of the constant theoretical value C2 p→3d = 16 Mb eV given by Thole and van der Laan for the series Ca (Z = 20) through Cu (Z = 29) [59]. The result is plotted in Fig. 10.15 as black circles.

Fig. 10.15 Radial transition matrix elements R in [nm] for the 2 p → 3d (L-shell, black circles) and 3 p → 3d (M-shell, blue squares) excitations in Ca (Z = 20) to Cu (Z = 29). The L-shell matrix elements were calculated according to (10.62) with ω taken as the 2 p3/2 binding energies with the theoretical value C2 p→3d = 16 Mb eV of Thole and van der Laan [59]. The M shell matrix elements were obtained from Hartree–Fock calculations for single charge cations by Hansen et al. [55]

522

10 Quantum Theory of X-Ray Absorption Spectroscopy

We also show in Fig. 10.15 as blue squares the Z dependence of the related 3 p → 3d (M-edge) radial matrix element. It was obtained from Hartree–Fock calculations by Hansen et al. [55].10 The 3 p → 3d radial matrix element shows the same trend with Z but is larger by a factor of about 6 than the 2 p → 3d one, owing to the greater overlap of core and valence states, as expected.

10.5.4 L-Shell XAS Intensities and Valence Shell Occupation According to (10.61) the integrated intensity of the L-shell XAS intensities scales with the number of empty 3d states or holes. With increasing Z from Ca (Z =20) to Zn (Z = 30) the total number of 3d spin orbitals, 2(2L + 1) = 10, is increasingly filled so that the white line intensity should gradually descrease. This is indeed observed as illustrated in Fig. 10.16a where we show the measured XAS spectra of the 3d metals [60], normalized for comparison to the same value above the resonances.

10

d

100

Number of d-electrons per atom 4 5 9 2 1 0 6 3 8 7 d d d d d d d d d d

(b) metals Ti

80

5

L3

Ti

L2

V 4

V

3d metals

Cr

3

Fe Co Ni

2 1

In te g r a te d L 3 + L 2 in te n s ity ( a .u .)

Normalized XAS intensity (a.u.)

(a)

Cr

60

Fe

40

Co Ni

20

Cu

0 100

continuum

0

(c) atoms

Ti+

80

Fe+

60

Cr+

V+

+ Sc

V+

40

0

10 20 Relative photon energy (eV)

30 20

Cu+

0

0

+ Ni

+ Co

2

4

6

8

Number of d-holes per atom

10

Fig. 10.16 a XAS absorption spectra of the 3d transition metals Ti–Ni [60], normalized to the same continuum jump above the edges and an energy scale relative to the inflection point of the absorption onset. The spectra reveal the change of the L3 –L2 spin-orbit splitting and the change in the total resonance “white-line” L3 +L2 intensity. b Total resonance intensities versus number of 3d holes in metals and c in individual trapped cations [56]. Also see Fig. 10.5 in [61] 10

We have converted the radial matrix element given in [55] in atomic units and multiplied by the √ √ reduced angular matrix element L = 2 to correspond to our convention (10.46).

10.5 Resonant XAS in Experiment and Theory

523

In Fig. 10.16b we also show the change of the integrated L3 plus L2 “white line” intensities with the number of valence holes [31] for the elementary metals and in (c) for cations in the form of singly ionized atoms [56]. The same linear dependence was also reported in [61] using electron energy loss spectroscopy. It is apparent from Fig. 10.16a that the intensity ratio between the L3 and L2 resonances changes significantly between Ni and Ti. In a one-electron model the L3 /L2 intensity ratio is 2/1, reflecting the relative degeneracy 2 j + 1 = 4 for j = 3/2 (L3 ) and 2 j + 1 = 2 for j = 1/2 (L2 ). The deviation was theoretically shown to be due to multi-electron correlation effects [59, 62] that shuffle intensities between the two resonances. This demonstrates the inadequacy of the one-electron theory. Remarkably, the sum of the L3 and L2 resonance intensities appears to be proportional to the number of 3d holes, in both one-electron and multi-electron theories. For XAS spectra of solids, the quantitative determination of the integrated white line intensities is complicated by an underlying background, which is typically modeled as the dashed step-like curve in Fig. 10.16a. This complicates the determination of the factors C2 p→3d , defined in (10.61). The value C2 p→3d  11 Mb eV determined from XAS and XMCD data for Fe, Co, and Ni metals, for example, where the underlying step intensity was subtracted [31, 63], is smaller than the value C2 p→3d = 16 Mb eV obtained from atomic calculations by Thole and van der Laan [59]. This may indicate that some of the resonant atomic oscillator strength is transferred into the continuum “background”.

10.5.5 Resonant Lineshapes in Atoms and Solids Theoretically, the absorption cross section is given by (10.60) and the resonances therefore should have the natural FWHM Lorentzian linewidth  listed in Table 10.2 and plotted in Fig. 10.4. In atoms the resonant widths beautifully exhibit the natural Lorentzian shape as seen for Ne in Fig. 10.10. In contrast the lineshapes and width in solids typically deviate significantly from Lorentzian forms. In the early days this could be attributed to limited Gaussian instrumental energy resolution, but modern spectra recorded with instrumental resolution well below the natural lifetime width still exhibit broader lines. This is revealed, for example, by the spectra in Fig. 10.16. In this case, experimentally observed widths are broadened by the spread of the 3d valence states caused by the band structure. As a consequence, the natural 2 p → 3d Lorenzian lineshape is transformed into a broader peak of reduced intensity, with preservation of the integrated resonance cross section. The broadening of resonances in solids is illustrated for the L3 resonances for the elemental metals Fe, Co, and Ni in Fig. 10.17. The experimental XAS cross sections shown as black curves were measured with linear polarization and polycrystalline samples with random magnetic domains to eliminate any polarization dependence. We have subtracted a small pre-edge background of about 0.2 Mb due to outer shell contributions, which underlies the resonant structure as shown for Fe in Fig. 7.14. The measured lineshapes are best accounted for by a fit with Voigt lineshapes shown in red which consist of Lorentzian and Gaus-

524

10 Quantum Theory of X-Ray Absorption Spectroscopy

sian components indicated on the right. The Voigt function is a convolution of the theoretical Lorentzian profile of natural linewidth , and a Gaussian, whose FWHM G and peak experimental cross section, σ0 , are determined by fit of the Voigt profile (see Appendix A.2.3)  ! ! 2 2 a(i(ω−E 0 −i/2) √ 0 )+/2) Re exp − a (ω−E erfc 2 2G 2 G exp ! ! , σXAS (ω) = σ0 a2 2 a exp 82 erfc 2√2 

(10.63)

G

G

√ where a = 2 ln 4 = 2.355 links the Gaussian FWHM, G , and rms width, σG , by G = aσG . The red experimental Voigt profiles may then be converted into the shown theoretical blue Lorentzian profiles under the assumption that the areas under the curves are preserved. The peak values of the experimental and converted theoretical cross sections are seen to differ significantly, and so do the widths of the associated Voigt and Lorentzian lineshapes. Both the red Voigt and blue Lorentzian resonances in Fig.

30

(a) Fe L3

theo 0

experiment Lorentzian =0.36 eV Voigt V =2.1 eV

(Lorentzian)

20

10

=0.36eV &

exp 0

6.9 Mb

G

=1.90eV)

area: 16.5 Mb eV

experiment Voigt fit

Absorption cross section (Mb)

Fig. 10.17 Experimental L3 absorption cross sections for Fe, Co, and Ni metals (black curves) as a function of photon energy. In all cases a small background of about 0.2 Mb resulting from absorption contributions from outer shells has been subtracted. Superimposed in red are Voigt lineshape fits consisting of the listed Lorentzian components of FWHM, , and Gaussian components of FWHM, G , that are listed on the right. Shown in blue are Lorentzians of natural linewidth  whose integrated cross sections are equal to those of the red Voigt lineshapes

0 704 705 706 707 708 709 710 15

(b) Co L3 experiment Lorentzian =0.43 eV Voigt V =1.64 eV

10 6.25 Mb

=0.43eV &

G

=1.40eV)

area:12.1 Mb eV

5 0 775 776 777 778 779 780 8

(c) Ni L3 6

experiment Lorentzian =0.48 eV Voigt V =1.09 eV

4.35 Mb

4

=0.48eV &

G

=0.80eV)

area: 6.07 Mb eV

2 0 851

852

853

854

Photon energy (eV)

855

10.5 Resonant XAS in Experiment and Theory

525

Table 10.5 Parameters for the L3 resonances of Fe, Co, and Ni metals. Listed are the atomic number densities ρa , the resonance energies E0 and wavelengths λ0 , the number of 3d valence shell exp holes Nh and electrons Ne , and the polarization averaged peak XAS cross section σ0 ρa Fe Co Ni

atoms nm3

eV

λ0 nm

84.9 90.9 91.4

707 778 853

1.75 1.59 1.45

E0

Nh

Ne

σ0 Mb

exp

x meV

exp

σ0theo Mb

xtheo meV

 meV

3.5 2.5 1.5

6.5 7.5 8.5

6.9 6.25 4.35

0.26 0.34 0.32

29.2 18.0 8.02

1.08 0.96 0.575

360 430 480

exp

The listed dipole transition widths x are obtained from the experimental cross sections by use of (10.55). The listed peak cross sections σ0theo correspond to the blue curves in Fig. 10.17 with the assumption that all oscillator strength is contained in a Lorentzian of FWHM  listed in the last column. The values xtheo correspond to these cross sections. The theoretical values may be taken as those of free individual atoms

10.17 for a given element have the same integrated L3 resonance values, which are 16.5 Mb eV (Fe), 12.1 Mb eV (Co), and 6.1 Mb eV (Ni). One may associate the blue theoretical cross sections with those of free individual atoms. The key parameters are summarized in Table 10.5 which lists the atomic number densities ρa , the resonance energies and wavelengths, the number of 3d valence shell holes Nh and electrons Ne , and the polarization averaged peak XAS cross section exp σ0 , corresponding to the peak values of the black or red curves in Fig. 10.17) [64]. exp By use of (10.55) they yield the listed dipolar matrix elements x . Similarly, the peak Lorentzian cross sections σ0theo of the blue curves correspond to the theoretical values xtheo . The last column lists the total natural (dipole plus Auger) decay widths  from Table 10.2.

10.5.6 Dipole Matrix Element, Oscillator Strength, and Sum Rules The energy integrated resonant absorption cross section is related to the quantum mechanical oscillator strength g of a resonance introduced in Sect. 6.4. For a dipole transition a → b of energy Eb − Ea = E0 , it is defined including polarization as [7] p

ga→b =

2m e E0 p |b|r ·  p |a|2 = −gb→a . 2

(10.64)

The energy integrated resonance cross section is related to the dimensionless oscillator strength by ∞ p

σXAS (ω) d(ω) = 0

2π 2 2 αf p λ2 p ga→b =  , m 2 ab  e  C

(10.65)

526

10 Quantum Theory of X-Ray Absorption Spectroscopy

where C = π h c r0 = 109.89 Mb eV. The last expression linking the oscillator p strength g p to the transition width ab only holds for a resonance, while the link of the integrated cross section to the oscillator strength is more general. It also holds for the non-resonant case as discussed by Egerton [65]. By integrating the absorption cross section up to a certain energy above the absorption edge one obtains a g value that represents the energy dependent fraction of the total oscillator strength of the shell. As an example, we consider the oscillator strength contained in the 3 p K-shell Rydberg resonance of Ne (see Fig. 10.10). By use of (10.65) and the value 1s→3 p = 7.6 × 10−2 meV at λ = 1.43 nm, we find an oscillator strength of g1s→3 p =

λ2 1s→3 p = 7.1 × 10−3 . 2C

(10.66)

Thus of the maximum oscillator strength of 2 for the two electrons in the K-shell, only a fraction of about 0.35% is contained in the lowest K-shell Rydberg resonance. Another example is the “white line” L3 resonance in Fe metal at λ = 1.75 nm with a value 2 p3/2 →3d = 1.08 meV (see Table 10.5), which corresponds to an oscillator strength of g2 p3/2 →3d =

λ2 2 p3/2 →3d = 0.15 2C

(10.67)

which is about 3.8% of the maximum oscillator strength 4 for the four electrons in the 2 p3/2 shell. Our value may be compared to the experimental value g2 p→3d = 0.14 per d valence hole obtained for single Mn5+ ions (3d 2 or 8 d valence holes) [66]. To do so, we account for the 3.5 d valence holes in our case (see Fig. 10.16) and for the missing L2 contribution by a factor of 3/2. This gives g2 p→3d = 0.064 per 3d valence hole for Fe metal which is smaller than the value for free Mn ions. This again suggests that in the metal some of the resonant oscillator strength is transferred to the non-resonant continuum.

10.5.6.1

Shell-Specific Versus Thomas-Reiche-Kuhn Sum Rule

The integrated L-shell white line cross section given by (10.61) or ∞ σXAS dω = C2 p→3d Nh

(10.68)

0

links the area under the resonant 2 p-shell (6 electrons) absorption cross section to the number of empty states or holes Nh in the 3d shell. The energy integrated resonant L-shell cross section is a fraction of the total integrated resonant plus continuum cross section which enters into the Thomas-Reiche-Kuhn (TRK) sum rule (6.73) and

10.6 Multi-electron Formalism: Multiplet Structure

527

links it to the total number of atomic electrons Z according to ∞ tot σXAS dω = C Z .

(10.69)

0

The TRK sum rule was previously discussed in Sect. 6.8 in conjunction with Fig. 6.11. If the electrons in all shells had unit oscillator strengths, the summed oscillator strengths would naturally be the number of electrons. According to (10.64) we would then have the oscillator strength per electron state |a to all final states |b

ga→b =

b

2m e E0 b

2

|b|r |a|2 = 1,

(10.70)

and the resonant dipole matrix element given by (10.40) per electron would be

b

ab =

(ω)2 8π 2 αf E0 2 = α . f λ2 2m e E0 m e c2

(10.71)

This beautifully simple result expresses the resonant dipole matrix element per electron in terms of the dimensionless fine structure constant, representing the photonelectron coupling, and the ratio of the square of the x-ray energy and the rest mass energy of the electron m e c2 = 511 keV. Figure 6.11b shows that, in practice, the black TRK sum rule curve lies below the red curve at all energies and catches up only later. The gradual increase with photon energy reveals that most of the oscillator strength is not contained in discrete resonances at the excitation thresholds but rather extends into the continuum, in good accord with our earlier discussion.

10.6 Multi-electron Formalism: Multiplet Structure In general, the one-electron and configuration pictures are not equivalent because the latter includes all possible couplings between angular momenta in the ground and excited states. This leads to splittings of absorption resonances (and photoemission peaks) due to exchange and correlation effects, so-called multiplet structure, as extensively discussed by van der Laan and Thole [67, 68] and de Groot and Kotani [33, 69]. The description of the valence electron states in different models, such as density functional band theory, weak and strong ligand field theory, and molecular orbital theory, has been reviewed by Stöhr and Siegmann [31]. Multiplet splittings are typically encountered in the L absorption spectra of free transition metal atoms and ions [54–56, 70] and when transition metal ions are embedded in inorganic oxides and coordination compounds [33, 34, 59, 62, 67–

528

10 Quantum Theory of X-Ray Absorption Spectroscopy

69]. In the configuration picture, the XAS spectrum consists of transitions from the electronic ground configuration consisting of a filled 2 p 6 core shell and a partially empty 3d N shell with 10 − N holes to an excited configuration where one electron has been removed from the core shell and added to the 3d shell according to 2 p6 3d N → 2 p 5 3d N +1 .

Electron excitation picture:

(10.72)

If the 3d shell is more than half full, it simplifies things to use the concept of holes instead of electrons and taking care of signs arising from the opposite charge. For example, Hund’s third rule states that the spin-orbit interaction changes sign and the LF splitting is upside down. With these rules we can use instead of (10.72) the following hole excitation scheme Hole excitation picture:

2 p 0 3d 10−N → 2 p 1 3d 9−N .

(10.73)

In the configuration picture one distinguishes two limiting coupling schemes for the spin and angular momenta of the individual electrons which may be in the same shell or different shells. The Russell-Saunders (or L S) coupling scheme describes the case where the Coulomb and exchange interactions between the electrons are larger than the spin-orbit interaction. The total spin and orbital momenta, denoted by capital letters, are then defined as the vector sum " individual elec" of those of the trons, indicated by small letters, according to S = i si and L = i li . The possible multiplet configurations are denoted by term symbols 2S+1

LJ.

(10.74)

The values L = 0, 1, 2, 3, 4, . . . are denoted in spectroscopic notation as S, P, D, F, G, . . ., and the simplest two electron spin states are the well-known 2S + 1 = 1 (S = 0) singlet and 2S + 1 = 3 (S = 1) triplet states. The smaller spinorbit interaction couples the spin and orbital momenta according to J = L + S,

(10.75)

where the vector addition results in possible J values L − S ≤ J ≤ L + S. When the spin-orbit coupling is stronger than the Coulomb and exchange interactions, one uses the j j-coupling scheme, where the individual electron spins and angular momenta are coupled first according to ji = li + si and the total angular momentum is then given by

ji . (10.76) J= i

In the one-electron model there is no summation over i and there is no distinction between L S and j j coupling. For multi-electron systems, one typically encounters a situation in between the two limiting cases. In so-called intermediate coupling, one expresses electronic states

10.6 Multi-electron Formalism: Multiplet Structure

529

as a linear combination of those in either the L S or j j coupling scheme. The two schemes can be transformed into each other through vector coupling coefficients given by the Wigner 3 j, 6 j, and 9 j symbols [9, 11, 33, 68], today implemented in different programming languages.11

10.6.1 Evolution of One-Electron to Multiplet Theory To demonstrate the evolution of the one-electron to the multi-electron or multiplet theory, we discuss the L-edge spectra of transition metal ions with d-shell electron occupations 3d 9 and 3d 8 which are encountered for the Ni+ and Co+ ions. These fundamental cases illustrate both the essence of the multiplet theory and how it merges into the limiting case of the one-electron theory.

10.6.1.1

Ground State d 9

We start with the case in Fig. 10.5 and assume that the ground state has the configuration 1s 2 2s 2 2 p 6 3s 2 3 p 6 3d 9 4s 0 corresponding to Ni+ ions. All closed shells contribute zero angular momenta J = L = S = 0 so that the angular momenta in the ground configuration are determined by the nine electrons or the single hole in the 3d shell. For the single hole, the labels l, s, and j are also the term labels L , S, J and the lowest 2S+1 L J terms are 2 D5/2 and 2 D3/2 , as shown in Fig. 10.18a. Similarly, the possible states in the excited state configuration 2 p 5 3d 10 are 2 P3/2 and 2 P1/2 . The spin-orbit splitting in the ground and final states is obtained by use of the spin-orbit coupling constants in Table 9.1. The splitting is given by so = 3ξnl /2, and the 3d 9 values given in Table 9.1 yield those given in Fig. 10.18a. Since the ground state s–o splitting, so  0.15 eV is larger than the thermal energy at room temperature (kB T = 0.026 eV), only the 2 D5/2 will typically be occupied. This causes the absence of the 2 D5/2 →2 P1/2 transition which is forbidden by the dipole selection rule J = 0, ±1. In general the dipole selection rules are as follows.

11

The description of angular momentum coupling between electrons or holes goes back to a formalism developed by the mathematicians Clebsch and Gordan in the late 1800s for the coupling of two angular momenta. The Clebsch–Gordon formalism is today handled by Wigner’s equivalent tensor formalism (3 j symbols) developed in the 1930s (original unpublished, [71]) [9, 11]. The formalism was extended by Wigner and Racah [72, 73] to the coupling of three and four angular momenta, expressed by the 6 j and 9 j symbols, respectively [9, 11].

530

10 Quantum Theory of X-Ray Absorption Spectroscopy

Transitions between electronic states |S, L , J, M J  and |S  , L  , J  , M J  are governed by the dipole selection rules S = S  − S = 0 L = L  − L = 0, ±1 J = J  − J = 0, ±1 M J = M J − M J = p = 0, ±1.

(10.77)

where p denotes the x-ray polarization (angular momentum). The theory is beautifully verified by the absorption spectrum for trapped Ni+ ions shown in black in Fig. 10.18b recorded by Hirsch et al. [56]. In contrast, the Ni metal spectrum shown in red exhibits a pronounced L2 resonance. As discussed in Sect. 10.5.3, the formation of valence bands in Ni metal involves an energy of 3 eV which is considerably larger than the 3d spin-orbit splitting of 0.15 eV. Hence the ground state in Ni metal contains both 2 D5/2 and 2 D3/2 states.

(a)

1

2

so=17eV

L3

2

P3/2

L2 2

D3/2 D5/2

2 so=0.15eV

Rel. absorption intensity

P1/2

(b)

L3

0.8 0.6

L2

0.4

Ni metal

0.2

Ni + ions

0 840

850 860 870 Photon energy (eV)

880

Fig. 10.18 a L-edge transitions between initial and final states for a Ni+ 2 p 6 3d 9 4s 0 ground state configuration. Both initial and final states are spin-orbit split and at room temperature only the lowest 2D 2 2 5/2 state is occupied, so that the D5/2 → P1/2 (L2 edge) is dipole forbidden. b Absorption spectrum (black) for the case in (a) recorded with trapped Ni+ ions [56] (and K. Hirsch, private communication) shifted by +3.5 eV to match the peak position of the Ni metal spectrum shown in red [74]. In the latter the L2 edge is observed due to band mixing of the 2 D5/2 and 2 D3/2 initial states

10.6 Multi-electron Formalism: Multiplet Structure

10.6.1.2

531

Ground State d 8

As expected, for the one-hole case, the multiplet theory gives the same result as the one-hole theory. The difference between the two treatment emerges when considering the case of two valence holes, i.e. the case of the ground electron state 2 p 6 3d 8 . The final state then has the 2 p 5 3d 9 electron or 2 p 3d hole configuration. The hole picture is thus considerably simpler, and for the 3d 2 → 2 p 3d hole transition we need to take into account the coupling or correlation effects between two holes in both the ground and excited states. For our discussion we follow Fig. 10.19 and first consider the ground state with two 3d holes. In L S coupling we have five possible states with term values {3 F, 3 P, 1 G, 1 D, 1 S} [33]. According to Hund’s rules, the ground state is 3 F. Hund’s rules [75] state as follows. For a given set of possible multiplets, the ground state is characterized by, (1) the maximum value of the total spin S allowed by the exclusion principle, and (2) the maximum value of the orbital angular momentum L consistent with the value of S. (3) The value of the total angular momentum J for the lowest state is equal to |L − S| when the shell is less than half full and L + S when the shell is more than half full. When the shell is half full, the application of the first rule gives L = 0, so that J = S. Also, for a set of possible multiplets, the term with the highest in S, and among those the highest in L, has the lowest energy. Generally, states of higher multiplicity have lower energies. Similarly, the possible L S combinations for the final states (one d hole and one p hole) are {3 F, 3 D, 3 P, 1 F, 1 D, 1 P}. The L S coupling terms for the ground and excited states resulting from the Coulomb and exchange interactions are illustrated in Fig. 10.19, which also gives their energetic separations calculated by Hartree–Fock theory [33, 59, 62, 67]. If the s–o coupling is weak relative to the Coulomb and exchange interaction, it leads to states 2S+1 L J with L − S ≤ J ≤ L + S and the allowed J states are split according to Eso (J ) =

ξnl [J (J + 1) − L(L + 1) − S(S + 1)] . 2

(10.78)

This leads to the splitting of the ground states illustrated in Fig. 10.19. At room temperature only the lowest state 3 F4 is populated, reducing the number of allowed transitions. The effect of the s–o interaction on the excited states is more complicated since the 2 p s–o interaction is larger (ξ2 p  10 eV in Table 9.1) than the exchange and Coulomb interaction ( 5 eV) and mixes different L S terms, so that L and S cease to be good quantum numbers. This is illustrated in Fig. 10.19, where the final states that

532

10 Quantum Theory of X-Ray Absorption Spectroscopy

2 p 53d 9 final state 15

Energy (eV)

~ S =0

3

3

2

3

0.65 F 3 - 0.60 F 3 - 0.45 D 3

10

1

3

spin-orbit splitting ~15eV

P

D

S =1

5 1

3

3

3

3

1

790

theoretical Co+ XAS spectrum 785

-0.75 F 3 - 0.47 F 3 - 0.45 D 3

F

-0.76 D 3 + 0.63 F 3 + 0.06 F

1

D

0

3

3

Energy (eV)

6

F4

S =0

S = L = J = MJ =

0

G

3

2 1

1

1

0

S

~

3

D

P

S =1

L3

780

3d 8 ground state 1

L2

3

Photon energy (eV)

P F

0

795

3

1

1

5

1

0.2

0.4 0.6 Intensity

0.8

1

0 0, +1 Dipole 0, +1 - transitions 0, +1 -

3 3

F

Coulomb & exchange (L-S) coupling ~5 eV

F2 F3 3 F4

3

spin-orbit splitting ~0.5 eV

Fig. 10.19 Multiplet terms and splitting for a transition metal ion with a 2 p 6 3d 8 electronic ground state and a 2 p 5 d 9 final state. The level splitting evolves from left to right through the Coulomb and exchange interaction, yielding the terms 2S+1 L, which in the presence of s–o interaction results in terms 2S+1 L J . In the ground state J remains a good quantum number because the spin-orbit interaction in the 3d shell is small, while the large s–o interaction in the 2 p core shell mixes 2S+1 L J terms [33]. The mixed final states that can be reached by dipole transitions from the 3 F4 ground state are listed (F. de Groot, private communication). The resulting transitions and the calculated absorption spectrum are shown on the right

can be reached by dipole transitions from the 3 F4 ground state are listed. They consist of linear combinations of different L S-terms of the same J .12 The transitions allowed by the dipole selection rule are indicated in blue, and the calculated absorption 12

The simplest s–o coupling of two angular momenta is the one-electron case discussed in the previous section, resulting in the splitting of the angular momentum l = 1 through the spin angular momentum s = ±1/2 into j = l + s, ( p3/2 ) and j = l − s, ( p1/2 ) states. The resulting m j spinorbit substates can then be written as linear combinations of the angular (spherical harmonics) and

10.6 Multi-electron Formalism: Multiplet Structure

8

Absorption cross section (Mb)

Fig. 10.20 Evolution of Co L-shell resonances from Co atoms deposited on a K film in a d 8 configuration to those in Co metal with an approximate d 7.5 configuration where the multiplet fine structure is no longer observed due to band formation [77] (and P. Gambardella private comm.)

533

L3

Co L resonances Co atoms on K film

6

Co metal

4

L2 2

0 775

780

785

790

795

800

Photon energy (eV) spectrum is shown on the right. The theory beautifully explains the spectra observed for individual trapped Co+ ions [56] and Co atoms isolated as impurities on alkali films [76, 77]. This is illustrated in Fig. 10.20, where we show the difference in the L-edge resonances observed for Co atoms on a K film (red) and those in bulk Co metal (black) [77].

10.6.1.3

Evolution from L S to j j Coupling for d 8

The complete transition from L S coupling to j j coupling for the d 8 ground state has been calculated by van der Laan and Thole [62] as shown in Fig. 10.21. As indicated by arrows on the right, the L S coupling spectrum gradually evolves into the j j coupling one with increasing s–o coupling and decreasing Coulomb and exchange coupling.13 Here the evolution of the spectrum is shown between two extreme cases. If the Coulomb and exchange interactions are zero, the spectrum (top trace) shows oneelectron like behavior with two peaks separated by the 2 p s–o splitting. In the other extreme of zero 2 p core s–o splitting but strong Coulomb and exchange splitting, the spectrum (bottom trace) is that calculated in pure L–S coupling. It consists of two peaks, corresponding to the dipole allowed 3 F → 3 F and 3 F → 3 D transitions. In the intermediate region, where both the 2 p core spin-orbit and 2 p–3d Coulomb and exchange splitting are present, a more complicated L-edge spectrum is found [59, 62]. the spin states, with the expansion coefficients given by the Clebsch–Gordon or Wigner 3 j coupling coefficients given in Table 10.3. 13 For a given electronic configuration, the states for each J block can be transformed between L S and j j coupling through 9 j recoupling coefficients (see (10.5) in [78]).

534

10 Quantum Theory of X-Ray Absorption Spectroscopy

Fig. 10.21 Calculated L-edge spectra for 2 p 6 3d 8 ground state configuration with a 3F4 ground state and a 2 p 5 3d 9 electronic final state configuration [59, 62] as shown in Fig. 10.19. The horizontal scale corresponds to excitation energy, while the 2 p–3d Coulomb and exchange interaction U ( p, d) and the 2 p spin-orbit coupling ξ2 p are varied vertically. The top trace corresponds to pure 2 p spinorbit coupling (U ( p, d) = 0) and the lowest trace to pure L–S coupling (ξ2 p = 0)

The above discussion has been limited to spectra of atoms not subjected to any anisotropic electric or magnetic interactions. In the absence of an exchange or external magnetic field, the degeneracy of the 3 F4 ground state in the M J substates −J ≤ M J ≤ J is not lifted and the XAS spectrum is independent of polarization since the 3 F4 state is spherically symmetric. In the presence of a strong magnetic field, an unequal thermal population of the M J substates in the the 3 F4 groundstate will cause a polarization dependence due to the x-ray magnetic circular (XMCD) and linear (XMLD) dichroism effects. In the presence of (electrostatic) ligand fields of lower than cubic symmetry, the valence charge density of the ground state will become anisotropic, leading to a polarization dependence due to the x-ray natural linear dichroism (XNLD) effect. These effects will be discussed in the following chapter.

References 1. D.C. Koningsberger, E.R. Prins, X-Ray Absorption: Principles, Applications, Techniques of EXAFS, SEXAFS and XANES (Wiley, New York, 1988) 2. J.J. Rehr, R.C. Albers, Rev. Mod. Phys. 72, 621 (2000) 3. G. Bunker, Introduction to XAFS: A Practical Guide to X-ray Absorption Fine Structure Spectroscopy (Cambridge University Press, Cambridge, 2010) 4. J.W. Cooper, Phys. Rev. 128, 681 (1962) 5. S.T. Manson, J.W. Cooper, Phys. Rev. 165, 126 (1968) 6. U. Fano, J.W. Cooper, Rev. Mod. Phys. 40, 441 (1968)

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47. M.A. Leutenegger et al., Phys. Rev. Lett. 125, 243–001 (2020) 48. M. Larsson, P. Baltzer, S. Svensson, B. Wannberg, N. Mårtensson, A.N. de Brito, N. Correia, M.P. Keanej, M. Carlsson-Göthe, L. Karlsson, J. Phys. B 23, 1175 (1990) 49. V.N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, L. Patthey, J. Synchrotron Rad. 17, 631 (2010) 50. F. Siegrist et al., Nature 571, 240 (2019) 51. M. Hofherr et al., Sci. Adv. 6, 8717 (2020) 52. E.J. McGuire, Phys. Rev. A 5, 1043 (1972) 53. W. Bambynek, B. Crasemann, R.W. Fink, H.U. Freund, H. Mark, C.D. Swift, R.E. Price, P.V. Rao, Rev. Mod. Phys. 44, 716 (1972) 54. M. Martins, K. Godehusen, T. Richter, P. Wernet, P. Zimmermann, J. Phys. B: At. Mol. Opt. Phys. 39, R79 (2006) 55. J.E. Hansen, H. Kjeldsen, F. Folkmann, M. Martins, J.B. West, J. Phys. B: At. Mol. Opt. Phys. 40, 293 (2007) 56. K. Hirsch, V. Zamudio-Bayer, F. Ameseder, A. Langenberg, J. Rittmann, M. Vogel, T. Möoller, B. v. Issendorff, J.T. Lau: Phys. Rev. A 85, 062–501 (2012) 57. A. Nilsson, N.Mårtensson: Physica B 208/209, 19 (1995) 58. A. Nilsson, J. Stöhr, T. Wiell, M. Aldén, P. Bennich, N. Wassdahl, M. Samant, S.S.P. Parkin, N. Mårtensson, J. Nordgren, B. Johansson, H.L. Skriver, Phys. Rev. B 54, 2917 (1996) 59. B.T. Thole, G. van der Laan, Phys. Rev. B 38, 3158 (1988) 60. Andreas Scherz: Ph.D. dissertation Freie Universiät (Berlin, 2004). Available on website http:// www-ssrl.slac.stanford.edu/stohr 61. J. Graetz, C.C. Ahn, H. Ouyang, P. Rez, B. Fultz, Phys. Rev. B 69, 235–103 (2004) 62. G. van der Laan, B.T. Thole, Phys. Rev. Lett. 60, 1977 (1988) 63. J. Stöhr, R. Nakajima, IBM J. Res. Develop. 42, 73 (1998) 64. J. Stöhr, A. Scherz, Phys. Rev. Lett. 115, 107–402 (2015) 65. R.F. Egerton, Ultramicroscopy 50, 13 (1993) 66. C. Blancard, D. Cubaynes, S. Guilbaud, J.M. Bizau, Phys. Rev. A 96, 013–410 (2017) 67. G. van der Laan, B.T. Thole, Phys. Rev. B 43, 13–401 (1991) 68. G. van der Laan: Hitchhiker’s guide to multiplet calculations, in Magnetism: A Synchrotron Radiation Approach, ed. by E. Beaurepaire, H. Bulou, F. Scheurer, J.P. Kappler (Springer, Berlin, 2006), p. 143 69. F. de Groot, Coord. Chem. Rev. 249, 31 (2005) 70. K. Hirsch, J.T. Lau, P.Klar, A. Langenberg, J. Probst, J. Rittmann, M. Vogel, V. Zamudio-Bayer, T. Möller, B. v. Issendorff: J. Phys. B: At. Mol. Opt. Phys 42, 154–029 (2009) 71. E. P. Wigner, On the matrices which reduce the Kronecker products of representation of S.R. groups, ed. by L.C. Biedenharn, H. van Dam. Quantum Theory of Angular Momentum. (Acad. Press, 1965), pp. 87–133 72. G. Racah, Phys. Rev. 62, 438 (1942) 73. G. Racah, Phys. Rev. 63, 367 (1943) 74. R. Nakajima, J. Stöhr, Y. Idzerda, Phys. Rev. B 59, 6421 (1999) 75. F. Hund, Z. Phys. 51, 759 (1928) 76. P. Gambardella, S.S. Dhesi, S. Gardonio, C. Grazioli, P. Ohresser, C. Carbone, Phys. Rev. Lett. 88, 047–202 (2002) 77. H. Brune, P. Gambardella, Surf. Sci. 603, 1812 (2009) 78. K.T. Moore, G. van der Laan, Rev. Mod. Phys. 81, 235 (2009)

Chapter 11

Quantum Theory of X-Ray Dichroism

11.1 Overview This chapter is devoted to the quantum formulation of the phenomenon of x-ray dichroism, i.e. the polarization dependence of resonant core to valence transitions. In quantum theory, the four types of polarization dependent absorption effects, XNLD, XMLD, XNCD, and XMCD, whose historical development has been outlined in Sect. 7.9 (see Fig. 7.18), arise from non-spherically symmetric charge or spin densities in the atomic volume. Owing to the tunability of photon energy and polarization at modern x-ray sources, dichroic studies have evolved into important and in many cases unique applications of x-rays. Here we treat this topic with emphasis on fundamental concepts that are illustrated by experimental results chosen to complement those in previous books [1–3]. The breath of dichroism is illustrated by applications ranging from surface and polymer science to spin dependent effects associated with localized spins on atoms as well as delocalized itinerant spins. XMCD studies of the behavior of itinerant spins are particularly challenging since they involve very small transient (pulsed) signals in nanostructured samples, giving rise to phenomena such as spin exchange scattering underlying giant magnetoresistance (GMR) or the creation of spin polarization through spin-orbit torques in the spin Hall effect (SHE).

11.2 Introduction to the Quantum Theory of Dichroism Quantum mechanically, the dichroic response arises from asymmetries in the polarization dependent transition matrix elements. The “magnetic” XMLD and XMCD effects are due to the spin-orbit interaction that couples the spin and charge degrees of freedom, while the “natural” XNLD and XNCD effects arise from anisotropies or spatial twists of the atomic charge density induced by bonding.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_11

537

538

11 Quantum Theory of X-Ray Dichroism

Valence charge and x-ray dichroism Linear dichroism (a)

x

z

jz

+

+

k

Circular dichroism

(b) j =3/2, jz= 1/2

pz

y

Intensity =0

XNLD

(d) Y1,1 L

+ -

- +

-

E Intensity =1

(c) a pz+ b dxy

k

E

k

Intensity =4

Intensity =1

XMLD

+

E

XNCD

E Intensity =1 Intensity =0

XMCD

Fig. 11.1 Four important types of dichroism, illustrated for transitions from spherically symmetric s core states (not shown) to p valence states. The illustration shows specific valence states that give rise to the four prominent types of dichroism indicated in the Figure. The anisotropy of the charge arises either from bonding alone (natural dichroism) or is induced by the existence of spin alignment (magnetic dichroism) in the presence of an external magnetic field or an internal exchange field which is sensed by the photon E-field through the spin-orbit coupling of the atomic charge

In a molecular orbital picture, the fundamental origin of the four effects may be envisioned as illustrated in Fig. 11.1 for the simplest case of K-shell transitions from a spherically symmetric s core shell to a p valence shell. XNLD measures the spatial anisotropy of the p valence shell, illustrated for the pz orbital in Fig. 11.1a. XMLD, shown in Fig. 11.1b, also measures the spatial anisotropy of the p shell but in the presence of spin alignment and the spin-orbit interaction. This is illustrated for a simple j = 3/2, jz = m j = 1/2 state, formed by vector coupling of l = 1 and s = 1/2, with spin alignment along the z axis (see Table 10.3). Note that for XMLD, the maximum orbital amplitude does not necessarily lie parallel (or perpendicular) to the spin direction. The origins of the circular dichroism effects are shown in Fig. 11.1c and d, using the distinction of angular momentum and helicity of circularly polarized light in Fig. 3.4. As shown in (c), XNCD may arise from the spatial handedness or chirality induced into the p valence shell by mixing with d orbitals when the ligand field lacks inversion symmetry. Then a pi orbital can mix with a di orbital, as illustrated for the pz + dx y orbital. The resulting orbital has opposite chirality along the x and y directions and no chirality along z. The transition intensity differs if the spatial rotation direction of the E-vector for circularly polarized light is the same or opposite to the spatial twist of the chiral valence orbital. The effect does not exist in the dipole approximation which neglects any k-dependent spatial rotation of the field as it traverses the atomic volume.

11.2 Introduction to the Quantum Theory of Dichroism

539

In contrast, the XMCD effect arises from the temporal rotation of the EM field which gives circular light its angular momentum, as shown in Fig. 11.1d. While the XNCD effect is due to the lack of spatial inversion symmetry, XMCD is due to the lack of time reversal symmetry. This is illustrated through the Y1,1 orbital of a p valence shell which possesses an angular momentum  in the direction shown in Fig. 11.1d. In this case the transition intensity to the Y1,1 orbital depends on the relative alignment of angular momentum of the circularly polarized light (rotation in time at the same point), and the angular momentum of the valence charge. The effect is opposite for transitions to the Y1,−1 orbital. The semi-classical and quantum descriptions of light interactions with anisotropic materials give the same answers. The quantum description is more fundamental and, in principle, does not require the use of semi-empirical data obtained by experiment. For example, the semi-classical theory describes charge asymmetries through a dielectric tensor whose components are determined semi-empirically. For x-rays this semi-empirical dielectric tensor is replaced by a quadrupolar tensor that reflects the anisotropy of the valence charge within the atomic volume. The expectation value of the quadrupole operator calculated by use of quantum mechanical wavefunctions then directly expresses various dichroic charge effects. This formulation may be extended to include the anisotropy of spin densities through the inclusion of spin-orbit coupling. In the following sections we illustrate the quantum mechanical calculation of dichroic effects, with emphasis on the XNLD and XMCD effects due to their practical importance in studies of chemical bonding and magnetism. More generally, we need to denote the polarization dependent dipole operators not just by the polarization label p but an additional Cartesian coordinate label α = x, y, z determined by the sample. For linearly polarized light α specifies the E direction relative to the charge distribution in the sample and for circularly polarized light indicates the propagation direction k relative to the magnetization direction of the sample. The completely defined polarization operators are then given by Pαp =  αp · r ,

(11.1)

where p = 0, ±1 (abbreviated p = 0, +, −) defines the photon angular momentum and α = x, y, z defines a coordinate direction. The so-defined operators are listed in Table 11.1. The sums over p in the previously stated sum rules (10.48)–(10.51) and (10.53) specifically assumed α = z. More generally, the polarization sums or averages can equivalently be expressed by use of any three orthogonal operators in Table 11.1. For a given direction α there are three possible angular momenta p = 0, ±1, and for a given p there are three directions α = x, y, z.

540

11 Quantum Theory of X-Ray Dichroism (l)

p

Table 11.1 Polarization dependent dipole operators Pα , expressed in terms of Racah tensors Cm where α indicates the direction of k or E and p = 0, +1, −1 characterizes the photon angular momentum Linear Polarization Ex :

Px0 =

x

Ey:

Py0 =

y

Ez:

Pz0 =

z

1 = r √ 2 i = r √ 2

 

(1)

(1)

(1)

(1)

C−1 − C+1 C−1 + C+1

 

(1)

= r C0

Circular Polarization kx:

ky:

kz:

1 Px+ = − √ (y + iz) 2 1 − Px = √ (y − iz) 2 1 + Py = − √ (z + ix) 2 1 − Py = √ (z − ix) 2 1 + Pz = − √ (x + iy) 2 1 Pz− = √ (x − iy) 2

 i (1) i  (1) (1) = − r √ C0 − r C−1 + C+1 2 2  i (1) i  (1) (1) = − r √ C0 + r C−1 + C+1 2 2  1 (1) i  (1) (1) = − r √ C0 − r C−1 − C+1 2 2  1 (1) i  (1) (1) = r √ C0 − r C−1 − C+1 2 2 (1)

= r C+1

(1) = r C−1

11.3 X-Ray Natural Linear Dichroism—XNLD We start our discussion of the polarization dependent x-ray absorption intensity with the case of XNLD and assume a non-magnetic system to emphasize the pure charge character of the effect without complications due to spin. In Sect. 7.9.3 we have already mentioned the “search light effect” as an easy way to visualize the polarization dependence. This is the most common way to describe the XNLD effect, and we shall therefore start by deriving its physical origin. The polarization dependent x-ray absorption rate and cross section given by (10.38) are determined by the dipole matrix elements (10.45) involving the polarization operators in Table 11.1. For linear polarization with E along α = x, y or z we can write the cross section of a transition from the core shell with angular momenta c, m to an unfilled valence shell with angular momenta L , M. Expressing the matrix elements (10.45) in terms of the linear ( p = 0) polarization operators Pα0 (α = x, y, x) in Table 11.1, the cross section 10.39 for linear polarization takes the form

11.3 X-Ray Natural Linear Dichroism—XNLD α,0 σabs

541

2  8π ωαf (/2)2 Pα0 2 = R  L , M| |c, m , 2 2  (ω−E 0 ) +(/2) r    m,M    A

(11.2)

|b| α0 ·r/r |a|2

where the underbrackets identify a polarization independent term A and the polarization dependent dipole matrix element in the form (11.1), where we have used (10.47) to factor the sum and squared matrix element. By use of Table 11.1, this expression may be written as α,0 =A σabs

 2 

 p (1) L, M c, m , e C α p m,M

(11.3)

p=0,±1

√ p where the coefficients eα are those in Table 11.1, i.e. ex± = ∓1/ 2, ex0 = 0, e± y = √ p i/ 2, e0y = 0, and ez± = 0, ez0 = 1 and fulfill the sum rule p |eα |2 = 1. To picture the polarization dependent transition intensities, we can express the transition probabilities from spherically symmetric core states to the real p and d molecular orbitals, whose wavefunctions are listed in Table 10.4 in terms of their linear combinations of complex spherical harmonics. Their charge densities, corresponding to the absolute values squared of the wavefunctions, are illustrated in Fig. 10.9. Also shown in that figure is how the anisotropic charge densities of the individual p and d orbitals become spherically symmetric upon their combinations. For the d orbitals the t2g and eg irreducible representations that are energetically separated in cubic symmetry furthermore consist of balanced in-plane (x−y) and out-of-plane (z) charge densities whose average is spherically symmetric.

11.3.1 The Search Light Effect According to (11.2) the angular dependence of the absorption cross section is determined by the absolute value squared of the core to valence transition matrix elements. For the most important K-edge and L-edge excitations, the transitions of interest are the 1s → 2 pi and 2 p → ndi (n = 3, 4, 5) transitions. In both cases the core shells are spherically symmetric as illustrated in Fig. 10.9. The polarization dependence then arises entirely from the anisotropic charge density of the individual pi and di valence orbitals. The angular dependence of the transition probabilities, which may be derived from the matrix elements in Fig. 10.6 (also see Fig. A.8), can be pictured along different E-vector directions relative to the orientation of the p and d orbital charge densities as shown in Fig. 11.2. For all 1s → 2 pi , (i = x, y, z) transitions we have a transition intensity 1/3 for E  x, y, z, respectively, and zero otherwise. The polarization averaged intensity per p orbital is L/(2L + 1) = 1/3, and the total 1s → 2 p intensity is L = 1 per spin in agreement with (10.48). For the L-edge (2 p → 3d) we have the polarization

542

11 Quantum Theory of X-Ray Dichroism

Relative transition probabilities for linear polarization 2p

1s

px py pz

dxy dxz dyz

dx2-y2

1 3

3 15

0

+

-

+

+ 0

4 15

-

-

3 15

+

3 15

magic angle o 54.5

0

+

+ 3 15

-

d3z2- r 2 z

0

0

3d

2p

x

1 15

y

1 15

+

Fig. 11.2 Polarization dependent transition probabilities per spin |b| α0 · r/r |a|2 (α = x, y or z) defined in (11.2) for the case of p and d orbitals. The listed relative intensities along the three coordinate axes are those for the E-vector aligned along the axes. The intensities are proportional to the orbital densities along the E direction, expressed by the search light effect discussed in the text. For 1s → 2 p K-shell excitation the transition probabilities are zero if E lies in the shown nodal plane, indicated in red. For the 2 p → d L-shell transitions, the excitation probabilities vanish if E lies along a nodal direction indicated in red

averaged intensity per d orbital L/(2L + 1) = 2/5 and the total 2 p → 3d intensity is L = 2 per spin in agreement with (10.48). The d3z 2 −r 2 orbital shown on the right has nodal directions at the “magic angle”, 54.5◦ from the z axis. In all cases, the summed polarization dependent transition probabilities in Fig. 11.2 follow the sum rules (10.48)–(10.51). For the K-edge, we readily recognize the foundation of the search light effect [1] which explains the polarization dependence in the following simple picture. The photoelectron is ejected from the spherically symmetric core state along the direction of the E-vector. The E-vector acts as a “search light” that senses the hole density of the valence orbital. The transition probability scales directly with the orbital density along E. For K-shell excitations the transition probability (intensity) vanishes when E lies in the nodal plane, as indicated in red on the left of Fig. 11.2. For L-edges the transition intensity is zero if the E-vector lies along a nodal axis, also indicated in red in the figure. In Fig. 11.3 we illustrate a strong search light polarization effect for two examples. In (a) we show XNLD spectra for oriented benzene (C6 H6 ) monolayers on a Ag(110) surface [4], and in (b) for Cu-phthalocyanine (Cu-Pc) monolayers adsorbed on Ag(100) [5, 6]. In both cases the molecules are bonded with their planes parallel to the surface. For benzene the empty π ∗ carbon orbitals are perpendicular and the σ ∗ orbitals parallel to the ring plane, while in Cu-Pc the unfilled dx 2 −y 2 Cu orbital (see Fig. 11.4 below) is parallel to the molecular plane. According to the search light effect, transition intensities are maximized when the E polarization vector points along the unfilled valence orbitals. The small residual L3 intensity in the red spec-

11.3 X-Ray Natural Linear Dichroism—XNLD

543

X-ray natural linear dichroism

*

(a)

C6H6 /Ag(110)

1

C C

C C C

* out-of-plane

E in-plane

N

N N Cu N

dx 2- y 2

Relative absorption intensity

C K-edge C

0.5

*

0 275

285

1.5 (b)

295

315

305

325

L3 Cu-phthalocyanine/Ag(100) Cu L-edges

1

0.5

L2

0 930

940

950

960

Photon Energy (eV)

Fig. 11.3 Examples of large contrast XNLD spectra and illustration of the search light effect. a K-shell spectra of benzene (C6 H6 ) chemisorbed in a lying-down geometry on Ag(110) [4] (also see Fig. 1.19), showing the linear dichroism associated with 1s → 2 p transitions for empty π ∗ and σ ∗ orbitals of the C atoms. b L-shell spectra of Cu-phthalocyanine also bonded lying down on Ag(110) [5, 6], revealing the strong dichroism of 2 p → 3d transitions to the half filled dx 2 −y 2 orbital on Cu atoms (see Fig. 11.4 below)

trum in Fig. 11.3b is due to the fact that the E-vector was actually at a finite angle of 20◦ from the normal of the Cu-Pc plane. The XNLD effect is similar to that observed in the layered transition metal compound La1.85 Sr 0.15 CuO4 [7] previously shown in Fig. 7.18.

11.3.2 XNLD and the Quadrupolar Valence Charge Density The absorption cross section for linear ( p = 0) polarization (11.3) expresses the dipole matrix elements in the one-electron form, in terms of transitions from core states, |c, m, to valence states |L , M. The anisotropy arises from that of the valence states. One can replace the transition matrix element by an expression for the anisotropic charge density of the valence shell, alone. It is expressed in terms of its deviation from a constant spherically symmetric charge distribution, which corresponds to the quadrupole moment of the charge density. The derivation of this result utilizes

544

11 Quantum Theory of X-Ray Dichroism

(b) Cu d orbital LF splitting

(a) Cu-phthalocyanine

dx 2- y 2

eg

Cu

C H N

3d

0eV

1=

-0.4eV

2=

-2.0eV

3=

-2.3eV

9

t2g

y

x

d3z 2 - r 2

0=

dxy dxz dyz

Fig. 11.4 a Structural model and coordinate axes of Cu-phthalocyanine. b Ligand field splitting of its 3d orbitals [5], where the z coordinate axis is taken along the normal of the molecular x − y plane in (a)

angular momentum sum rules which underlie the derivation of the famous XMCD sum rules [8–13]. The key step in the derivation is to rewrite the squared matrix element so that the final state that contains the core hole can be eliminated by use of quantum mechanical closure relations according to  m

2 |L , M|C (1) p |c, m| =



(1) ∗ L , M|C (1) p |c, mL , M|C p |c, m

m

= (−1) p L , M|C (1) p

 m



|c, mc, m| C−(1)p |L , M. (11.4) 



projection

The projection summation does not yield unity, but it is related to the unit closure relation by an L-dependent expression, which allows its elimination as discussed in Appendix A.8.3. This leaves matrix elements of a product of first order ten(1) sors C (1) p C − p that connect only valence states. The first order tensor product can be expressed in terms of second order tensors C L(2) by the relations given in Table A.3, which in turn define the components of the quadrupole tensor of the charge. It has a particularly simple principal axis form, given by,1 1

The diagonal quadrupole tensor corresponds to the tensor order parameter or Saupe matrix [14– 16] discussed in Sect. 7.10.1, utilized in NEXAFS studies of the orientation of nematic liquid crystals and polymer surfaces [17].

11.3 X-Ray Natural Linear Dichroism—XNLD

Qxx Q yy Q zz

545

  3 (2) = 1 − 3 sin θ cos φ = − C2(2) + C−2 , 2

  3 (2) = 1 − 3 sin2 θ sin2 φ = C0(2) + C2(2) + C−2 , 2 = 1 − 3 cos2 θ = − 2 C0(2) , 2

C0(2)

2

(11.5) (11.6) (11.7)

with a vanishing trace, Q x x + Q yy + Q zz = 0. After some lengthy algebra one obtains the following alternative expression for the polarization dependent XAS cross section [10, 12, 13] α,0 =A σabs

  L 2L + 3 1−  φ L |Q αα |φ L  . 6L + 3 2L

(11.8)

For the K- and L-edges, the valence states |φ L  with angular momentum L are the p or d orbitals in Table 10.4 or their linear combinations. This leads to the pictorial description shown in Fig. 11.5. The vanishing trace of Q αα is seen to be fulfilled by adding the three values along the orthogonal axes in the figure. An average over α then yields the cross section and matrix elements given by (10.55) and (10.56), and the angle and energy integrated x-ray absorption cross section are determined by the radial transition matrix element Rc→L and the number of unoccupied valence states Nh . The quadrupolar formulation shows that the core hole state is eliminated by the closure relation and the polarization dependence is entirely determined by the charge distribution of the valence orbitals.

Quadrupole matrix elements 1s

2p

p x py pz 4 -5

x 2 5

< for linear polarization

4 7

d 3z 2 - r 2 4 -7

+

+ 2 -7

+ 4 7

3d

2 -7

2 5

-

i

2p

dx2 - y 2

y

i| Q

dxy dxz dyz

z

+




+

| Lz=2

>

When an atom is embedded into a solid, the understanding of the existence of orbital moments is more complicated. The importance of orbital contributions to the magnetic properties of transition metal ions in insulators was thoroughly studied by means of electron paramagnetic resonance (EPR) in the 1960s as reviewed by Abragam and Bleaney [36]. Orbital contributions also played an important role in unraveling the localized versus delocalized behavior of transition metal impurities in metallic hosts, which in the 1960s and 1970s formed one of the key topics in condensed matter physics, associated with buzz words like Anderson localization and the Kondo effect as reviewed by Hirst [37]. If we picture the bonding between atoms through the formation of molecular orbitals shown in Fig. 10.9, one finds that all individual orbitals have zero orbital moments. This is readily verified by calculating the expectation value oi |Lz |oi  for any of the orbitals in Table 10.4. One may picture the destruction of the atomic orbital moments through bonding (i.e. the formation of molecular orbitals) as illustrated in Fig. 11.11b for the specific case of the dx y orbital and an external H-field along z. An orbiting electron on the central atom will experience a Coulomb attraction near the corners of the bonding square where the positive neighbor ions are located, and the orbit is broken up. In textbooks, this is referred as “quenching” of orbital moments. In solids, the ligand field (LF) splitting of the different orbitals is typically of order of a few eV. If we now turn on the intra-atomic s–o interaction in the valence shell, which for transition metal atoms is of order ξ3d  50 meV, one creates small orbital moments due to the mixing of LF orbitals. This is illustrated in Fig. 11.12, where we ↑ ↑ show the effect of mixing the LF orbitals dx y and dx 2 −y 2 by the s–o interaction. For simplicity, we only show the s–o mixing of orbitals of the same spin (↑= +sz ) and ignore the mixing of states of up and down spin, discussed later in Sect. 11.6. The perturbed ligand field orbitals are given by

558

11 Quantum Theory of X-Ray Dichroism

Fig. 11.12 Creation of orbital momenta Lz  by the spin-orbit (s–o) interaction. We assume spin-up ↑ ↑ alignment by an external field H  z. The pure spin-up ligand field orbitals dx y and dx 2 −y 2 have

↑ ↑ no orbital momenta, but they are induced in the d˜x y and d˜x 2 −y 2 orbitals, given by (11.20), through the s–o interaction. The orbitals possess angular momenta Lz  given by (11.21), as shown. On the right we indicate the spin-orbit-induced orbital rotation in the x−y plane around H  z by assuming ξ/ = 0.2

|d˜x↑y  = |dx↑y  + ↑

↑

↑

↑

dx 2 −y 2 |ξ Lz sz |dx y 

|d˜x 2 −y 2  = |dx 2 −y 2  +

Edx y − Edx 2 −y2 ↑

↑

dx y |ξ Lz sz |dx 2 −y 2  Edx 2 −y2 − Edx y

↑

|dx 2 −y 2  = |dx↑y  − i

ξ ↑ |d 2 2  x −y

↑

|dx↑y  = |dx 2 −y 2  − i

ξ |d ↑  (11.20) xy

where we have used Edx y −Edx 2 −y2 = , and the spin and angular momentum operator rules in Tables A.6 and A.7, namely ↑ |sz | ↑ = 1/2, and Lz |dx y  = −2i|dx 2 −y 2 , and Lz |dx 2 −y 2  = 2i|dx y  to obtain the expressions on the right. The s–o interaction shifts the upper orbital up in energy by 2ξ 2 / and the lower orbital down by −2ξ 2 / , and the perturbed orbitals have angular momenta Lz  given by 

  4ξ  4ξ d˜x y |Lz | d˜x y = , and d˜x 2 −y 2 |Lz | d˜x 2 −y 2 = − .

(11.21)

These are the values listed in Fig. 11.12. In the 3d magnetic transition metals Fe, Co and Ni, for example, one finds orbital moments that are only about 5–10% of the size of spin moments [2, 13, 38, 39]. On the right of Fig. 11.12 we also illustrate how one may picture the creation of an orbital moment along z as arising from the s–o induced charge rotation in the x−y plane.

11.5 X-Ray Magnetic Circular Dichroism—XMCD

559

In the following we shall show how spin and orbital magnetic moments can be determined by XMCD through measured resonance intensities. The key to the link of the magnetic properties of matter and XMCD intensities are the famous XMCD sum rules which we will discuss next. We start with the sum rule for the determination of the orbital magnetic moment, first derived by Thole et al. [8]. It is more robust and more easily derived than the sum rule associated with the spin magnetic moment [9], discussed later.

11.5.2 XMCD Sum Rule for the Orbital Moment The XMCD effect only exists indirectly through the coupling of the E-vector to the charge which in turn is coupled to the spin through the spin-orbit interaction. We are interested in the determination of the spin and orbital magnetic moments, either of a ferromagnetic sample whose domains are aligned by an external magnetic field (typically 2 T), or a paramagnetic sample where a net spin orientation is induced by application of a large magnetic field (of order of 10 T). In the latter case one furthermore has to assure that the Zeeman splitting by the applied magnetic field is larger than the thermal energy so that such experiments are carried out at low temperature (of order of 1 K) [40]. For our discussion we consider a valence shell of angular momentum L (e.g. L = 2 for the 3d bands) and consider the band model illustrated in Fig. 11.10c. For convenience, we fix the sample magnetization direction m so that the XMCD measurements are performed by switching the incident circular polarization. We consider k-summed band states, projected on their spin sz and orbital M quantum numbers, of the form ↑

↓

↑

↓

|φ L M (E) = |φ L M (E) + |φ L M (E) = c M (E) |L M ↑ + e M (E) |L M ↓. (11.22) ↑

↓

The coefficients c M (E) and e M (E) deviate from unity through mixing of states by the ligand field and s–o interactions. We have expressed the angular band states in a spherical harmonics basis |L , M = |Y L ,M , but one may also use a molecular orbital (MO) basis since the MOs are also orthogonal. When done so, the band model in the limit of no k dispersion (flat bands) and no exchange interaction reduces to a LF model. The orbital momentum of the valence states at a given energy E is given by Lz (E) =

 M

φ L M (E)|Lz |φ L M (E) = 



 ↑ ↓ |c M (E)|2 +|e M (E)|2 M. (11.23)

M

In the absence of the s–o interaction the sum over M vanishes. When integrated over filled electron states or empty hole states, we obtain the total orbital momentum of the valence shell according to

560

11 Quantum Theory of X-Ray Dichroism

EF

∞ Lz (E) dE =

Lz  = − −∞

Lz (E) dE.

(11.24)

EF

Owing to the opposite sign of electron and hole states, Lz  vanishes when the integration is performed over the entire band, which corresponds to the shell being completely empty or filled. The orbital moment therefore depends on the band filling. It is this quantity, Lz , which we want to determine as the difference of resonant x-ray absorption cross sections measured with opposite circular polarizations. In XMCD, the valence shell L is picked out by the dominant c → L = c+1 dipole transition channel (see Fig. 10.2). Sensitivity to the orbital momentum of the valence shell arises from its coupling to the photon angular momentum of the circularly polarized x-rays. XMCD measures the hole part of the valence shell angular momentum since only transitions to empty states are allowed. The orbital moment sum rule is determined by the total contributions of all electrons in a given core shell of angular momentum c. The core s–o splitting separates the c − sz and c + sz (e.g. p1/2 and p3/2 ) states which differ only in the spin but not orbital contributions. While sensitivity to the spin magnetic moment therefore requires the separation of the intensities of the transition from the two s–o core states (see below), this is not required for the determination of the orbital moment of the valence shell. This fact makes the orbital moment determination of the valence shell more robust, and it can be measured even if the core shell s–o splitting is small so that the XAS resonances overlap or if there is no core s–o splitting as for the K-shell. We consider circularly polarized x-rays incident parallel to the z axis of our coordinate system, described by dipole operators Pz± = rC±(1) in Table 11.1, with the sample magnetization also along z as in Fig. 11.10c. For our assumed geometry with k  z, the polarization dependent XAS cross section (10.39) p

p

σXAS =

λ2 ab (/2)2 π  (ω−E0 )2 +(/2)2

(11.25)

for circular polarization denoted by p = ± is specifically given by z,± = σXAS

2  8πωαf (/2)2 (1) R2 φ L ,M (E)|C± |c, m, sz  .(11.26) 2 2  (ω−E0 ) +(/2)    M,m,sz A

The underbracket identifies the polarization independent term A introduced in (11.2). The key to the orbital sum rule is the following expression for the difference in matrix elements in (11.26), which is proven by use of the matrix elements in Table 10.1 for the dominant c → c + 1 = L dipole transition (e.g. 2 p → 3d)

11.5 X-Ray Magnetic Circular Dichroism—XMCD



561

(1) |φ L M (E)|C1(1) |c, m, sz |2 − |φ L M (E)|C−1 |c, m, sz |2

M,m,sz

=

 1  ↑ 1 Lz (E) ↓ . |c M (E)|2 + |e M (E)|2 M = 2L + 1 M 2L + 1 

(11.27)

This allows us to write (11.26) in terms of the cross-section difference (dropping the label z) + − − σXAS =A σXAS

1 Lz (E) . 2L + 1 

(11.28)

We can eliminate the prefactor A by normalization to the polarization averaged cross section given by (10.55) or3 σXAS =

+ − 0 + σXAS + σXAS σXAS L Nh =A , 3 3(2L + 1)

(11.30)

where the polarization average follows from (10.53). This gives the following sum rule expression by integration over energy, first derived by Thole et al. [8].4 The orbital magnetic moment of a valence shell of angular momentum L containing Nh empty states is determined by the energy integrated resonant absorption cross section for a c → c+1 = L core to valence transition according to  + − σ − ) dω (σ mo Lz  = L Nh  + XAS − XAS 0 . =−  μB (σXAS + σXAS + σXAS ) dω   

(11.32)

from experiment

Here we have used (11.19) to link Lz  and the orbital magnetic moment m o . Application of the sum rule requires knowledge of the unfilled number of valence states, Nh .

3

For the 3d transition metals, one may rewrite (11.30) by use of 0 = σXAS

+ − + σXAS σXAS 2

(11.29)

res = σ 0 which is more easily determined experimentally. so that σXAS abs 4 For the transition c → c + 1 = L, the angular momentum prefactor in [8] reduces to

2L(L + 1) = −L c(c + 1) − L(L + 1) − 2 − + which together with their inverted sign σXAS − σXAS reduces to our expression.

(11.31)

562

11 Quantum Theory of X-Ray Dichroism

+ − We will see below that the XMCD cross-section difference σXAS − σXAS in the numerator eliminates any non-magnetic background. In contrast, the experimental + − 0 + σXAS + σXAS in the denomdetermination of the resonant cross-section sum σXAS inator is often complicated by the existence of a non-resonant background that needs to be subtracted. It is typically modeled as the dashed step-like curve in Fig. 10.16a, but this procedure may introduce sizable errors. It may be advantageous to use a different concept based on the transferability of the radial transition matrix element and another sum rule link of the polarization averaged “white line” intensity with the number of holes suggested in [41] and expressed by (11.30). For the L absorption edge the integral over energy in the denominator of (11.32) may then be replaced by use of (10.61) in the form

∞ ∞ + − 0 res σXAS + σXAS + σXAS dω = 3 σabs d(ω) 0

0

2 3Nh = 4π 2 ω αf R22 p→3d 15   

(11.33)

C2 p→3d

which yields the simpler expression, Lz  2 mo = =−  μB 3 C2 p→3d



+ − (σXAS − σXAS ) dω,   

(11.34)

from experiment

where C2 p→3d = 16 Mb eV is the theoretical value given by Thole and van der Laan for the series Ca (Z = 20) through Cu (Z = 29) [42]. The value C2 p→3d  11 Mb eV determined from XAS and XMCD data for Fe, Co, and Ni metals, for example, where the underlying step intensity was subtracted [2, 41], is smaller than the value 16 Mb eV obtained from atomic calculations by Thole and van der Laan [42].

11.5.3 Experimental Studies of Orbital Magnetism The capability of XMCD to directly determine orbital magnetic moments is unique. In the past their presence was only indirectly revealed by material dependent deviations of the gyromagnetic factor or g-factor from the pure spin value of g = 2 measured by magnetic resonance techniques [43, 44] or direction dependent differences in the saturation magnetization [45]. From the sum rule (11.32) we seethat the orbital moment vanishes if the energy + − − σXAS ) dω = 0. For the L absorption integrated difference in cross section (σXAS edges this means that the XMCD difference intensity vanishes when integrated over both the L3 and L2 resonances. In other words, the different intensities for the L3 and

11.5 X-Ray Magnetic Circular Dichroism—XMCD

563

L2 resonances need to have the same areas but opposite signs. A beautiful example of this case is given in Fig. 11.13a, where the Cu L-edge XMCD difference spectrum is shown for Cu that has become magnetic in an alloy with Co due to hybridization of the 3d valence states [46]. The L3 and L2 XMCD effects are seen to be opposite with approximately the same area under the curves. In this case the orbital moment vanishes because the 3d valence shell of Cu is nearly full. Figure 11.13b reveals how the orbital moment of Co changes due to its bonding environment. The spectra were recorded by Gambardella et al. [47] using electron yield detection with the sample at T = 10 K and magnetically saturated in an external magnetic field of B = 7 T. In order to emphasize the changes of the L3 and L2 XMCD intensities, the spectra have been normalized to the same L2 area. The relative increase of the L3 area is directly proportional to the increase in orbital moment which follows the intuitive model shown in Fig. 11.11. The LF induced quenching of the moment in bulk Co is increasingly removed by the decreasing number of bonds with neighbor

1 (a) 0

Cu

-1 920 2

Cu10Co90 alloy

930

(b) 0 Rel. XMCD intensity

L2

L3

940

L3

950

960

970

L2

Co

Co bulk

-2 Bulk -4

Co monolayer

Monolayer on Pt -6

Pt(111)

Co chains

-8 Atomic chains on Pt -10 770

Pt(997)

780 790 800 Photon energy (eV)

810

Fig. 11.13 a XMCD difference spectra of Cu atoms in a Co10 Co90 alloy [46], b XMCD signal for Co atoms in bulk films, monolayers on Pt(111), and chains on Pt(997) [47]

564

11 Quantum Theory of X-Ray Dichroism

Table 11.2 Orbital magnetic moments L z  measured by XMCD for Co atoms on Pt surfaces [47–49] System L z  [μB ] Co metal (hcp) CoPt (111) ML CoPt (997) ML CoPt (997) chain CoPt (111) adatom

0.14 0.29 0.31 0.68 ± 0.05 1.1 ± 0.1

atoms. The increase in orbital moments is quantified in Table 11.2 for Co on Pt surfaces [47, 48]. For the Co on Pt systems, the orbital moments were also linked with the magnetocrystalline anisotropy, which was directly determined from XMCD magnetization loops [47–49]. Finally we mention that enhanced orbital moments are also found in XMCD studies of gas phase atoms, dimers, and clusters [50, 51].

11.5.4 XMCD Sum Rule for the Spin Moment For the derivation of the spin sum rule, first derived by Carra et al. [9] for the general case of an atom in a ligand field that is subjected to a magnetic field, we again describe the valence shell by the band states (11.35). The spin magnetic moment defined by (11.18) as m s = −2 μB sz / is then theoretically defined by integrating the expression sz (E) =



φ L M (E)|sz |φ L M (E) =

M

   ↑ ↓ |c M (E)|2 −|e M (E)|2 2 M

(11.35)

over filled electron states or empty hole states according to EF sz  = − −∞

∞ sz (E) dE =

sz (E) dE.

(11.36)

EF

As for the orbital moment, given by (11.24), the spin moment vanishes when the integration is performed over the entire band and therefore depends on the band filling. In contrast to the orbital moment case, we need to separate the contributions from the two s–o split core states and calculate their cross sections separately. We again consider circularly polarized x-rays incident parallel to the z axis of our coordinate system, described by dipole operators Pz± = rC±(1) in Table 11.1, with the sample

11.5 X-Ray Magnetic Circular Dichroism—XMCD

565

magnetization also along z as in Fig. 11.10c. The cross section for the two core s–o substates j+ = c+sz and j− = c−sz is then given by rewriting (11.26) as ± [σabs ] j± = A

2  (1) φ L ,M (E)|C± | j± m j  ,

(11.37)

M,m,sz

where the states | j± m j  with −m j ≤ j± ≤ m j are written in terms of a linear combination of spherical harmonics as given in Table 10.3 for the p j± (c = 1) and d j± (c = 2) states. The matrix elements in (11.37) can be evaluated by use of Table 10.1 for the dominant c → c + 1 = L transitions (also see Figs. 10.6 and A.8). From an experimental point of view it is difficult (and not done in practice) to keep track of the directions of the sample magnetization and the x-ray angular momenta which determine the sign of the XMCD difference effect. Furthermore it is a nuisance that the spin angular momentum sz and the spin moment m s have opposite signs (see (11.18)). It is therefore customary to define the experimental XMCD difference intensity to be negative at the lower energy and more intense j+ resonance. This corresponds to the following theoretical sum rule link of the spin angular momentum sz and the XMCD cross-section difference.5 The spin angular momentum sz  of a valence shell of orbital angular momentum L containing Nh empty states may be obtained from the energy integrated resonant absorption cross sections of transitions from the separate s–o core states j+ = c+sz and j− = c−sz to the valence shell L = c + 1 according to 2sz  2(2 L +3) Tz  + = 3Nh  L 

 

+ − L j+(σXAS −σXAS ) dω − L−1  + − j+ + j− (σXAS + σXAS





+ − j−(σXAS −σXAS ) dω . 0 + σXAS ) dω



from experiment

(11.38) Here Tz  is the expectation value of the intra-atomic magnetic dipole operator along the external H-field direction z, given by Tα = [s − 3ˆr(ˆr · s]α =



Qαβ sβ ,

β

5

With L = c + 1 the angular momentum factors given in [9] reduce to L(L + 1) − 2 − c(c + 1) 2 = , and 3c 3 L(L + 1)[L(L + 1) + 2 c(c + 1) + 4] − 3(c − 1)2 (c + 2)2 6+4L = 6 c L(L + 1) 3L

.

(11.39)

566

11 Quantum Theory of X-Ray Dichroism

where Qαβ are the components of the quadrupole tensor whose diagonal elements are given by (11.5)–(11.7). Application of the sum rule requires knowledge of the unfilled number of valence states, Nh . We see that the determination of the spin angular momentum or magnetic moment linked by m s = −2sz μB / is complicated by the presence of the additional Tz  term in (11.38). In the following section we develop a physical picture of the meaning of our mathematical orbital and spin moment sum rule expressions by considering a specific example, which also serves to test the theoretical predictions through experimental results.

11.6 Test of the Sum Rules: Cu-Phthalocyanine For the exploration of the sum rule from a theoretical and experimental perspective, we discuss XMCD results for the paramagnetic molecule Cu-phthalocyanine (CuPc) whose dichroic properties have been studied by Stepanow et al. [5, 6]. We have already discussed the strongly anisotropic XNLD spectra of Cu-Pc adsorbed as a monolayer on Ag(100) in conjunction with Fig. 11.3. The XNLD spectra of the same sample are determined by transitions from the s–o split 2 p core states to the half empty 3dx 2 −y 2 orbital in the molecule adsorbed with its plane parallel to the Ag substrate surface. Below we consider the quantitative correlation predicted by the sum rules between the XMCD effect and the orbital and spin magnetic moments.

11.6.1 Electronic Structure of Cu-Pc Metal-Pcs, named after the Greek words for rock oil (naphtha) and blue (cyan) and nicknamed “magnetic blue”, are flat molecules with a central metal atom surrounded by a network of N, C, and H atoms as illustrated in Fig. 11.4a. Cu-Pc is a model spin 1/2 system with an 2 p 6 3d 9 ground state whose ligand field splitting of the Cu 3d orbitals is shown in Fig. 11.4b. The single unpaired hole in the 3dx 2 −y 2 orbital gives rise to paramagnetism, and spin alignment may be created at low temperature by application of a strong external magnetic field. The L-edge x-ray absorption transition diagram is illustrated in Fig. 11.14a, and the XMCD spectra recorded at normal incidence to the molecular plane, i.e. along z, in a field H  z of value B = μ0 H = 5 Tesla and a sample temperature of T = 6 K [5] are shown in Fig. 11.14b. The difference spectrum is shown in (c).

11.6 Test of the Sum Rules: Cu-Phthalocyanine

567

1.4

I+

(b)

(a)

3dx

L3

-y

2

L2

2

Relative intensity

1.2

Cu L-edge XMCD Cu-phthalocyanine

1 0.8

I-

0.6

k

H

N N Cu N N

0.4 0.2 0 0.4

2p3/2 so=20eV

2p1/2

(c) 0.2

L3

0

L2

-0.2 -0.4 -0.6 925

930

935 940 945 950 Photon energy (eV)

955

960

Fig. 11.14 a One-electron or hole model of 2 p3/2 and 2 p1/2 core to 3dx 2 −y 2 valence transitions in Cu-phthalocyanine. b XMCD spectra of Cu-phthalocyanine, recorded for k  H  z, as indicated, at a field strength B = μ0 H = 5 Tesla and a sample temperature of T = 6 K [5]

Below we shall discuss the origin of the spin and orbital XMCD effects and their angular dependence. For the Cu-Pc case this is facilitated by the simple 2 p 6 3d 9 electron or 2 p 0 3d 1 hole ground state, which can conveniently be treated in a onehole model shown in Fig. 11.14a. In this case the s–o effects, arising in the core and valence shell, can be separately accounted for and owing to the small s–o and Zeeman interactions in the 3d valence shell both interactions can be treated in perturbation. Our calculation below which treats the s–o coupling to first order in the coupling constant ξ will be seen to agree very well with the full calculation presented by Stepanow et al. [5]. Our treatment has the advantage of yielding analytical expressions that offer greater physical transparency. The key to our calculation is to include the additional effects of the s–o coupling and applied external field on the partially empty 3dx 2 −y 2 valence orbital which determines the electronic transitions as illustrated in Fig. 11.14a. Let us consider the relative size of the two interactions, which determines their perturbation treatment. If we ignore the s–o interaction, the 3dx 2 −y 2 ground state has pure spin-1/2 character and in a field B = μ0 H =10 Tesla exhibits a Zeeman splitting of E = 2μB H  1 meV. This Zeeman splitting may be compared to the s–o splitting of the unfilled 3dx 2 −y 2 orbital, which for the LF structure shown in Fig. 11.4b may be estimated following our previous discussion in conjunction with Fig. 11.12. We obtain a value of order

568

11 Quantum Theory of X-Ray Dichroism

E  ξ 2 /(E0 − E3 ) ∼ 5 meV. Hence both interactions are smaller by a factor of more than 100 relative to the size of the LF splitting of order of 2 eV and can well be treated as a perturbation. The tricky part of the s–o treatment arises when the external field is rotated. Then one needs to account for the rotated direction of the spin quantization axis relative to the LF coordinate system (x, y, z). This is most easily understood by assuming a ferromagnetic sample. In this case, spins are held parallel by the strong inter-atomic exchange field of order 103 −104 Tesla. The exchange field only acts on the spin system, leading to the splitting of the spin-up and-down states shown in Fig. 11.10c. Application of an external magnetic field Hext may simply be pictured as aligning different domains into a macroscopic magnetization or collective spin direction. The spin quantization direction is therefore naturally given by the direction of the externally applied field Hext . The effects of orbital fields due to the s–o interaction, which are largely quenched by the LF splitting, appear only in higher order. In paramagnets, of interest here, there is no exchange field and magnetic alignment is created only by an applied external magnetic field of order 1–10 Tesla, which acts on both spin and orbital moments. However, as in the ferromagnetic 3d transition metals, the spin moment also dominates in paramagnetic 3d compounds. One may therefore assume that to first order the spins are aligned parallel to the direction of the applied field Hext , so that the spin quantization axis may be taken along Hext . Spin rotations may be treated by use of Pauli’s quantum mechanical 2D spinor formalism [2, 52] or classical Euler rotations of 3D spin vectors. We shall briefly show the equivalence of the two treatments by considering the s–o interaction.

11.6.2 Treatment of the Spin-Orbit Interaction We first consider the case where the spin quantization axis is the same as the z axis of the (x, y, z) frame of the LF orbitals, defined in Fig. 11.4. In this frame the s–o interaction Hamiltonian (9.7) has the vector form   1 1 Hso = ξ(lx sx + l y s y + lz sz ) = ξ lz sz + l+ s− + l− s+ , 2 2

(11.40)

where the operators are related by [53] 1 2

i 2

1 2

i 2

lx = (l+ + l− ), l y = − (l+ − l− ), sx = (s+ + s− ), s y = − (s+ − s− )

(11.41)

l+ = lx + il y , l− = lx − il y , s+ = sx + is y , s− = sx − is y .

(11.42)

or

11.6 Test of the Sum Rules: Cu-Phthalocyanine

569

The matrix elements in the Cartesian (x, y, z)-basis are evaluated by use of Tables A.6 and A.7 in the Appendix or by the spherical basis expressions6 lz | , m =  m | , m  l± | , m =  ( + 1) − m(m ± 1) | , m ± 1 sz |s, m s  =  m s |s, m s   s± |s, m s  =  s(s + 1) − m s (m s ± 1) |s, m s ± 1.

(11.44) (11.45)

The s–o interaction couples the charge and spin degrees of freedom of the sample, and the form given by (11.40) is correct only if the z axis in the frame of the LF orbitals can also be taken as the spin quantization axis. By use of the operator rules (11.44) and (11.45), we can derive the wavefunctions of the perturbed unfilled state d˜x 2 −y 2 , proceeding similar to the calculation of the s–o perturbed states given by (11.20) in Fig. 11.4b. We obtain for H  s  z with the notation ↑= +sz and ↓= −sz   iξ ξ |dx↑y  + |dx↓z  − i|d yz↓  E0 −E2 2(E0 −E3 )   i ξ ξ s−z ↓ ↓ |d˜x 2 −y 2  = |d˜x 2 −y 2  = |dx 2 −y 2  − |dx↓y  − |dx↑z  + i|d yz↑  . E0 −E2 2(E0 −E3 ) (11.46) ↑

↑

|d˜x 2 −y 2  = |d˜x 2 −y 2  = |dx 2 −y 2  + s+z

By use of these functions and the lα (α = x, y, x) operator rules in Table A.7,7 we obtain the following expectation values for the orbital momenta along z 4ξ E0 −E2 4ξ ↓ ↓ d˜x 2 −y 2 |lz |d˜x 2 −y 2  = − . E0 −E2 ↑

↑

d˜x 2 −y 2 |lz |d˜x 2 −y 2  = +

(11.47)

For the states (11.46), corresponding to s  ±z, we obtain the orbital momentum expectation values lx  = l y  = 0.

6

For our s = 1/2 case we have  1 1 1 =± ± sz ± 2 2 2 

1 1 =1 + s+ − 2 2 

1 1 = 1. − s− + 2 2

7

In particular, lz |dx 2 −y 2  = 2i|dx y  and lz |dx y  = −2i|dx 2 −y 2 .

(11.43)

570

11.6.2.1

11 Quantum Theory of X-Ray Dichroism

Rotation of the Spin States in the Spinor Formalism

The LF states for spin alignment s  ±x and s  ±y are obtained by rotating the spin parts of the wavefunctions (11.46). To do so we associate the spin states along z with the spinor components according to     1 0 and ↓ = −sz = . ↑ = +sz = 0 1

(11.48)

In the Pauli spinor formalism the rotated spin states for s  ±x and s  ±y can then be expressed as a linear combination of the states for s  ±z as8     1 1 s±x = √ ↑ ± ↓ , and s±y = √ ↑ ± i ↓ . 2 2

(11.49)

Upon rotation of s  ±x, the states (11.46) become  1  ↑ s+x ↓ |d˜x 2 −y 2  = √ |d˜x 2 −y 2  + |d˜x 2 −y 2  2   1 s−x ↑ ↓ |d˜x 2 −y 2  = √ |d˜x 2 −y 2  − |d˜x 2 −y 2  2

(11.50) (11.51)

and the rotates states for s  ±y are given by  1  ↑ s+y ↓ |d˜x 2 −y 2  = √ |d˜x 2 −y 2  + i |d˜x 2 −y 2  2   1 s−y ↑ ↓ |d˜x 2 −y 2  = √ |d˜x 2 −y 2  − i |d˜x 2 −y 2  . 2

(11.52) (11.53)

We obtain the following orbital momentum expectation values lx  and l y  d˜x 2 −y 2 |lx |d˜x 2 −y 2  = d˜x 2 −y 2 |l y |d˜x 2 −y 2  = ± s±x

s±x

s±y

s±y

ξ . E0 −E3

(11.54)

As expected from Fig. 11.4a, we find the same expectation values along x and y because of the molecular C4v symmetry of Cu-Pc. This result compares to our earlier one for lz  given by (11.47) which in the same notation reads d˜x 2 −y 2 |lz |d˜x 2 −y 2  = ± s±z

8

See Sect. 8.4.2 in [2].

s±z

4ξ . E0 −E2

(11.55)

11.6 Test of the Sum Rules: Cu-Phthalocyanine

571

Comparison of the results reveals an anisotropic orbital angular momentum, with an approximately four times larger value in the direction normal to the molecular plan than in the molecular plane.

11.6.2.2

Vector Rotation of the S–O Hamiltonian

The above results for the anisotropy of lα  with α = x, y, z are a special case of an arbitrary orientation of the spin quantization axis relative to the LF frame. In the following we shall derive the same results by treating the rotation of the spin frame in a vector model through classical Euler angles. We then need to distinguish the Cartesian LF or charge frame x, y, z from the rotated spin frame which we shall denote as x, ˜ y˜ , z˜ , as illustrated in Fig. 11.15. The two frames are transformed into each ˜ Owing to the symmetry in the x − y molecular other through Euler angles θ˜ and φ. plane, we may take y˜ to lie in the x-y–plane, as shown, so that the (x, ˜ y˜ , z˜ ) frame is obtained from the (x, y, z) frame by a φ˜ rotation about z and then a θ˜ rotation about y˜ . Separation of the LF and spin frames allows us to write the s–o Hamiltonian for ˜ defined in Fig. 11.15. any Hext direction in terms of the Euler rotation angles θ˜ and φ, For example, by use of the relations given at the bottom of Fig. 11.15 we obtain the following expressions for Hext aligned along the directions x, y or z of the LF frame   Hext , s  z = z˜ : Hso = ξ lx sx˜ + l y s y˜ + lz sz˜   Hext , s  y = z˜ : Hso = ξ −lx s y˜ + l y sz˜ − lz sx˜   Hext , s  x = z˜ : Hso = ξ lx sz˜ + l y s y˜ − lz sx˜ .

(11.56) (11.57) (11.58)

We can now directly calculate the wavefunctions associated with these three forms of the s–o Hamiltonian. For (11.56) we have (x, ˜ y˜ , z˜ ) = (x, y, z) and the wavefunctions are given by (11.46) leading to the orbital momentum expectation values lz  given by (11.55). The expectation values l y  and lx  are calculated with the eigenfunctions of the Hamiltonians (11.57) and (11.58), respectively. For example, the eigenstate of (11.58) for s  +x, i.e. z˜ = x, is obtained with the notations ↑ = +sz˜ and ↓ = −sz˜ as s+x ↑ |d˜x 2 −y 2  = |dx 2 −y 2  −

  iξ ξ ↓ ↓ ↑ |d  + |d  − i|d yz  . E0 − E 2 x y 2(E0 −E3 ) x z

(11.59)

572

11 Quantum Theory of X-Ray Dichroism

Spin-orbit coupling of charge and spin frames

Hext z spin ~ frame z

dx -y 2

~

charge frame

2

y~ ~

x

~

~ x

y

Fig. 11.15 Relationship of the ligand field coordinate system of the sample (x, y, z) (black) used to describe the molecular orbitals and the rotated spin frame (x, ˜ y˜ , z˜ ) (blue), whose quantization direction z˜ is assumed to lie along the externally applied magnetic field Hext . For Cu-Pc, the directions of the x˜ and y˜ axes in the plane perpendicular to Hext can be arbitrarily chosen for symmetry reasons, and we take y˜ to lie in the x-y–plane, as shown. The spherical angles θ˜ and φ˜ then uniquely specify the direction of z˜ in the (x, y, z) frame. The (x, ˜ y˜ , z˜ ) frame is obtained from the (x, y, z) system by a φ˜ rotation about z and then a θ˜ rotation about y˜ . The relations between the spin coordinates in the two frames are given at the bottom

Similarly, we have for s  −x, i.e. z˜ = −x, ↓

|d˜x 2 −y 2  = |dx 2 −y 2  + s−x

  iξ ξ |dx↑y  − |dx↑z  + i|d yz↓  . E0 − E 2 2(E0 −E3 )

(11.60)

By use of these wavefunctions and those for the cases s  ±y, which we skip for brevity, we obtain the same orbital momentum expectation values given by (11.54).9

It may be surprising that the same angular momentum expectation values, e.g. lx , are calculated with the states (11.50) and (11.51), where ↑= +sz and ↓= −sz refer to the spin quantization axis z, and the seemingly different corresponding states (11.59) and (11.60), where ↑= +sz˜ and ↓= −sz˜ with the quantization axis defined by z˜ = x. This arises from the fact that in the calculation of lx  (and similarly for l y ) the spin parts of the states always satisfy the orthogonality relations ↑ | ↑ = ↓ | ↓ = 1 and ↑ | ↓ = ↓ | ↑ = 0. 9

11.6 Test of the Sum Rules: Cu-Phthalocyanine

573

11.6.3 Comparison of Orbital Momenta in Theory and Experiment sα The states |d˜x 2 −y 2  given by (11.46) for α = ±z and (11.59)–(11.60) for α = ±x consist of the zeroth order states |dx 2 −y 2  plus first order s–o perturbation contributions from other LF states. With increasing field strengths Hext  z˜ the spin degeneracy of the spin-up and-down states ±sz˜ will be lifted and with decreasing temperature the sample magnetization M will increase according to the Brillouin function

 M = Msat tanh

 μB (2sz˜  + l z˜ )Hext . kB T

(11.61)

This expression includes the contributions from both the spin and orbital momenta in the applied field direction z˜ . The dominant spin contribution is isotropic in first order, 2s = 1, while the orbital one differs according to (11.54) and (11.55). With the s–o parameter ξ = 102 meV (see Table 9.1) and the LF splittings E0 − E2 = 2.0 eV and E0 − E3 = 2.3 eV given in Fig. 11.4 we obtain the numerical values lz  =

4ξ ξ = 0.204, and lx  = = 0.044 E0 − E2 E0 − E3

(11.62)

In Fig. 11.16a we give our theoretical results for lz  and lx  resulting from the s–o mixing of the different di LF orbitals, assuming spin alignment by an applied field. ↑ The signs of the listed orbital momenta correspond to the spin-up |d˜x 2 −y 2  orbital. In Fig. 11.16b we show as blue squares the detailed angular dependence of the orbital momenta measured by Stepanow et al. [5], recorded at B = μ0 Hext = 5 Tesla and a sample temperature of T = 6 K. The solid blue curve corresponds to the theoretical expression l = k

  M lz  cos2 θ˜ + lx  sin2 θ˜ , Msat

(11.63)

where θ˜ is defined in Fig. 11.15. The curve was calculated with M/Msat = 0.58, reflecting the experimental parameters.10 We also accounted for electron delocalization that is not included in a simple LF model by use of a covalency reduction factor k = 0.9 [5]. We find excellent agreement between experiment and theory, thus illustrating the power of the angle-dependent XMCD orbital momentum sum rule.

10

This corresponds to a ground state splitting of E = μB (2s + l z )Hext  0.7 meV.

574

11 Quantum Theory of X-Ray Dichroism

(a) Cu d orbitals in Cu-phthalocyanine and l

dx 2- y 2 0=

d3z 2 - r 2 dxy dxz dyz

1=

0eV

H,s || x

lz

lx

4 ——— 0

-0.4eV

-

———

2

0

-

-2.3eV

4 0

-

3

——–––

0

1

2= -2.0eV 3=

H,s || z

-

3

filled orbitals

——– 2

2

0 2

-

3

-

1

3

-

0

3

-

3

(b) 0.12 Orbital momentum l (h)

Fig. 11.16 a Orbital momenta lα  in units of  of the LF states in Cu-Pc in the presence of the s–o interaction (11.40) for alignment of the H-field either along the surface normal z, lz , or parallel to the molecular plane, α = x or y, lx  = l y . The listed values for lα  are for saturation spin alignment by the applied H-field assuming that the spin-up state ↑ |d˜x 2 −y 2  is occupied. b Measured angle dependence of l at B = μ0 H = 5 Tesla at T = 6 K (squares) [5], and theory (blue line) according to (11.63)

Experiment

0.10

Theory

0.08 0.06 0.04 0.02 0.00

0

20 40 60 80 ~ Angle from surface normal z (deg.)

11.6.4 Spin Momenta in Theory and Experiment In analogy to the expressions for the angle dependent s–o Hamiltonian given by (11.56)–(11.58), the magnetic dipole operator is given by the following expressions for k, Hext , s  x, y or z, k, Hext , s  z : Tz = Q zx sx˜ + Q zy s y˜ + Q zz sz˜ k, Hext , s  y : T y = −Q yx s y˜ + Q yy sz˜ − Q yz sx˜ k, Hext , s  x : Tx = Q x x sz˜ + Q x y s y˜ − Q x z sx˜ .

(11.64) (11.65) (11.66)

Here Q i j are the elements of the quadrupole tensor introduced in Sect. 11.3.2 and the spin quantization axis is denoted as z˜ . The matrix elements are calculated by expressing the operators Q i j by spherical tensors as given in Table 11.3. By use of Table 11.3, we can rewrite (11.64)–(11.66) in the convenient operator forms

11.6 Test of the Sum Rules: Cu-Phthalocyanine

575

Table 11.3 Quadrupole operator elements expressed in terms of Racah’s spherical tensors       (2) (2) (2) (2) (2) Q x x = C0 − 23 C2 + C−2 Q x y = Q yx = i 23 C2 − C−2       (2) (2) (2) (2) (2) 3 C − C−1 Q yy = C0 + 23 C2 + C−2 Q x z = Q zx = 2  1  (2) (2) (2) Q zz = − 2 C0 Q yz = Q zy = − i 23 C1 + C−1

z˜  z :

Tz = −2 C0(2) sz˜   

−

3 (2) C s+ + 2 −1

3 (2) C s− 2 1

Q zz

z˜  y :

   3 (2) (2) (2) T y = C0 + C + C−2 sz˜ 2  2   Q yy



   3 3 (2) (2) (2) (2) (2) (2) (2) + +i C1(2) +i C−1 −C2 +C−2 +i C1 +i C−1 s+ + C2 −C−2 s− 8 8 z˜  x :

   3 (2) (2) (2) Tx = C 0 − C + C−2 sz˜ 2  2   Qxx



   3 (2) 3 (2) (2) (2) (2) (2) + −C1(2) +C−1 C2 −C−2 −C1 +C−1 s+ + −C2(2) +C−2 s− , 8 8 (11.67) where the operators s± refer to the spin frame (x, ˜ y˜ , z˜ ). The expectation values of Tα  are calculated with the wavefunctions for the corresponding field directions, which for α = z are given by (11.46) and for α = x by (11.59)–(11.60). In (11.67) we have indicated by underbrackets that Tα  exists even in the absence of the s–o interaction. Then the atomic spin density can simply rotate into the Hext field direction, and the spin-flip contributions represented by the operators s± are absent. The angular dependence of Tα  then simply reflects the anisotropy of the charge distribution in the atomic sphere, expressed by the diagonal elements of the quadrupole tensor, Qαα . In the middle column of Fig. 11.17a we show the calculated values of Tz  and T  = Tx  = T y  for the di orbitals in Cu-Pc in the absence of the s–o interaction but full spin alignment along Hext . Of particular interest is the dx 2 −y 2 orbital probed in XMCD, for which we also give the s–o induced change of the dipole moment ↑ calculated with the spin-up states (11.46) |d˜x 2 −y 2 sz and the states (11.59)–(11.60)

576

11 Quantum Theory of X-Ray Dichroism

(a) Cu d orbitals in Cu-phthalocyanine and T no s–o coupling

with s–o coupling

7 Tz 2

dx 2- y 2

0

0

dxz dyz

-

3

-2 1=

-0.4eV

1 2

dxy

3

-1+ ——–––

-1

0= 0eV

d3z 2 - r 2

2 - ——––– -

7 T||

2=

-2.0eV

3=

-2.3eV

filled orbitals -1

-2 1

(b) 2s + 7T

Momenta (h)

1.5

Experiment Theory

7T

1.0 0.5

2s

0.0 magic angle

-0.5 0

20 Angle

~

40

60

80

from surface normal z (deg.)

Fig. 11.17 a Orbital energies, calculated saturation values of 7Tα  for the di orbitals of Cu-Pc without s–o coupling, assuming spin alignment along H and a spin-up ground state. For the orbital ↑ |dx 2 −y 2  probed by XMCD we also give on the right the 7Tα  values in the presence of s–o coupling, given by 7Tz  = 2 − 3ξ/(E0 − E3 ) = 1.87 and 7T  = −1 + 3ξ/2(E0 − E3 ) = −0.93. b Comparison of the measured (blue squares) [5] and calculated (blue curve) angular dependence of the sum rule values according to (11.68) with M/Msat = 0.58. The decomposition into pure spin (black) and dipolar (red) contributions is also shown. Note that 7T = 0 near the magic angle of θ˜ = 54.7◦ ↑

|d˜x 2 −y 2 sx , respectively. On the right of Fig. 11.17a we show the additional s–o ↑ contribution for the |d˜x 2 −y 2  state. It is seen to scale with the ratio of the s–o coupling constant ξ and the LF splitting E0 − Ei , and we obtain 7Tz  = 2 − 3ξ/(E0 − E3 ) = 1.87 and 7T  = −1 + 3ξ/2(E0 − E3 ) = −0.93 at magnetic saturation. The angular dependence of the XMCD sum rule signal is expressed by 2s + 7T =

  M 2s + 7Tz  cos2 θ˜ + 2s + 7Tx  sin2 θ˜ , Msat

(11.68)

11.7 Application of XMCD to the Study of Transient Spin Effects

577

where θ˜ is defined in Fig. 11.15. The spin momentum 2s = 1 becomes anisotropic only in second order of ξ . In our first order treatment the total moment 2s + Lα  is anisotropic only due to the anisotropy of the orbital moment. This gives rise to the anisotropy of the g-factor measured in electron paramagnetic resonance [54]. The calculated angular dependence of the XMCD sum rule signal (11.68) for Cu-Pc is shown as a blue curve in Fig. 11.17, where it is compared to the measured values shown as blue squares, assuming the experimental value M/Msat = 0.58. In the figure we also show the theoretical decomposition into contributions of 2s (black line) and 7Tα  (red curve). For the case of Cu-Pc we find excellent agreement between the theoretical XMCD sum rules and experiment. By use of the Tα sum rule of Stöhr and König [10], one may eliminate the large angular anisotropy due to the 7Tα term by summing over three measurements with the circularly polarized light incident along the three orthogonal directions of Hext along x, y, z or a single “magic angle”, θ˜ = 54.7◦ , measurement. This is nicely revealed in Fig. 11.17, where the 7Tα term vanishes at θ˜  55◦ . In closing we note that Cu-Pc is a special case that can be described by oneelectron theory. The s–o coupling in the core and valence shell can then be treated separately, with the small s–o coupling in the valence shell treated by perturbation theory. The XMCD sum rules are valid also for the more complicated cases where the s–o interaction in the core and valence shell can no longer be treated separately, since in the final state configuration the coupling of the core hole and multi-hole valence states results in multiplet effects. These more complicated cases have been discussed in more detail in [55–59].

11.7 Application of XMCD to the Study of Transient Spin Effects Since its discovery by Schütz and collaborators [60] in 1987, XMCD spectroscopy has had significant impact in modern magnetism research, which around the same time was put on a new trajectory through the discovery of the giant magnetoresistance (GMR) effect independently discovered around 1988 by Albert Fert and collaborators [61] and Peter Grünberg and collaborators [62], who were awarded the Nobel Prize in Physics in 2007. Some of the impact of XMCD has been reviewed previously in [2, 63–66]. In contrast to bulk magnetic crystals, whose spin structure was largely solved by neutron scattering, modern applications of magnetism are based on artificially engineered thin films and nanostructures, utilized for writing and reading magnetic data bits in computers and servers. The power of XMCD for the investigation of such materials is founded on the following five x-ray capabilities. • The large x-ray intensity available at synchrotron and XFEL sources facilitates short measurement times. • The photon energy tunability allows the determination of atom and bond-specific contributions.

578

11 Quantum Theory of X-Ray Dichroism

• Polarization control coupled with large resonant cross sections allows separation of charge and spin effects with high sensitivity. • The short wavelength opens the door for imaging on the nanometer length scale, down to the atomic size. • The pulsed nature of synchrotron and FEL sources provides temporal resolution, down to the intrinsic femtosecond timescale of changes in the electronic and spin structure. Most applications of XMCD have been associated with picking apart the static magnetic properties of materials through spectroscopy and spatially resolved imaging of atom specific spin and orbital moments. The atomic moments associated with the valence electrons are either intrinsically created by the exchange interaction in magnetic systems, induced in non-magnetic atoms through hybridization with magnetic neighbors, or induced in paramagnets by application of external magnetic fields. The so-created static magnetic moments typically vary in size from 0.01 μB to a few μB [2, 63, 64]. In order to highlight other remarkable capabilities of XMCD we will present below its use to study the transient behavior of magnetic structures which underlies the function of magnetic devices. While atomic moments may be pictured as relatively large static spin imbalances localized within the atomic volume, the field of spin transport [67–69] is concerned with dynamic phenomena caused by application of a voltage. Spin currents may be created by preferential spin dependent exchange scattering when electrons flow through a ferromagnet or by spin-orbit scattering when a current flows through a heavy metal with large s–o coupling such as Pt. In the following we will present two state-of-the-art applications of XMCD that utilize all five x-ray capabilities outlined above.

11.7.1 Spin Accumulation in Cu upon Injection from Co Our first example is related to current flow through the nanostructure shown previously in Fig. 1.22a. As shown in that figure, a spin polarized current created in one ferromagnetic (FM) layer, after transmission through a non-magnetic Cu decoupling layer, may exert a torque on the static atomic moments in another FM layer and change its magnetization direction. In the following we will show that one may actually directly measure the sign and magnitude of the spin current that flows into the Cu spacer layer. Such an experiment requires “x-rays at their best” by combining spectroscopic capabilities with time-resolved nanoscale imaging as reviewed in [2, 65, 66]. While spin dependent current or resistivity effects are typically studied by transport measurements, XMCD can provide more detailed information on the spin dependent scattering mechanisms. For example, the GMR effect arises from the change in resistivity between parallel and antiparallel orientations of the static magnetization directions in two FM layers in Fig. 1.22a. The thin non-magnetic Cu layer serves to

11.7 Application of XMCD to the Study of Transient Spin Effects

579

decouple the magnetic orientations in the two FM layers so that they can be rotated relative to each other. The size of the GMR effect depends on the magnitude of the spin current in the Cu layer which, however, is not measured. Early XMCD measurements revealed the existence of magnetic moments in Cu interface atoms adjacent to magnetic layers such as Co and Fe [46, 70, 71]. This effect previously illustrated in Fig. 11.13 for Cu atoms in a Co10 Co90 alloy [46] arises from the hybridization of the 3d bands and is a static “proximity” effect. It exists without current flow. In the presence of current flow from a FM layer into Cu, an additional much smaller transient effect occurs. When a charge current flows through the FM, the traveling electrons may scatter on the magnetic atoms, and according to Fermi’s golden rule, the scattering cross section depends on the density of empty atomic states that the electrons can scatter into. But as illustrated in Fig. 11.10 there is an imbalance of localized empty spin-up and-down states. So naturally one kind of spin is scattered more. This argument hinges on the important fact that the spin is conserved in the scattering process, known as Mott’s two current model [2], developed in the 1930s [72–74]. The charge current which is the sum of the two spins may be broken up into the two spin channels. When the current in the ferromagnet is injected into a non-magnet, there is preferential accumulation of one of the spins at the interface. The spin imbalance slowly disappears with distance, called the spin diffusion length, in the non-magnetic layer. By keeping it sufficiently thin, the imbalance or spin accumulation may be observed, as shown for Co/Cu interface in a pillar in Fig. 11.18 [75]. The measurement combined XMCD transmission microscopy (see Sect. 8.2) with the synchronized detection of the effect of current pulses as schematically illustrated in Fig. 11.18a. While the number of photons in the individual 100 ps x-ray pulses separated by 2.1 ns varies slightly, the average number of photons emitted during the 780 ns storage ring repetition time is constant and is therefore best used to determine the tiny XMCD signal difference between current on/off periods. The sample structure and current flow direction is shown in Fig. 11.18b. Of interest is the transient XMCD signal in the Cu layer due to the injection of a spin current from the adjacent Co layer. Figure 11.18c shows the result for the Cu XMCD contrast, spatially resolved with a resolution of 35 nm, measured with opposite circular x-ray polarization directions and current density of 107 A/cm2 . It is interesting that such a large current density would result in melting of a typical Cu wire, but in the pillar configuration the large surface area allows sufficient heat dissipation. From the measurement, the transiently induced moment per Cu atom was derived to be 3 × 10−5 μB . The measurement also revealed a linear dependence of the transient moment with current and that the sign of the spin current in Cu is the same as that in Co on the other side of the interface. The measurement reveals a remarkable XMCD sensitivity equivalent to the magnetic moment of only about 50 Fe atoms. A related XMCD experiment also detected spin accumulation in Cu without flow of a charge current by use of “spin pumping”, consisting of the creation of precessing magnetic moments by microwaves in an adjacent Ni80 Fe20 permalloy layer [76]. On

580

11 Quantum Theory of X-Ray Dichroism

(a)

Current on

Current pulses even ring cycle, 780 ns

odd ring cycle, 780 ns

X-ray pulses

zone plate 35nm spot

avalanche photo diode

nanopillar sample

(b) electron flow

SiN

current density 7 2 10 A/cm

Ru M

Co

Cu

SiO2

5 mA

Au

XMCD intensity change

(c)

-

+8x10-5 Cu

L3-edge

+

+4x10-5 0 -5

- 4x10

-5

- 8x10

240nm Fig. 11.18 a Timing sequence and synchronization between current pulses sent through the nanopillar sample (shown enlarged in (b)) and individual x-ray pulses (100 ps wide, separated by 2.1 ns) emitted by different electron bunches within the total 780 ns cycle time of the storage ring. The monochromatic x-rays of 200 meV bandwidth, tuned to the Cu L3 resonance (933 eV), were focused by a zoneplate based transmission x-ray microscope to a 35 nm spot which was scanned across the sample, with the time and polarization dependent transmitted intensity measured by an avalanche photodiode [75]. b Schematic of the nanoscale pillar, with Ru and Au contacts, containing a Co based structure whose magnetization M was aligned along the pillar and a  30-nm-thick Cu layer into which the spin polarized electrons generated in the Co layer were injected. c Normalized transient XMCD signal (current on/off), arising from spin accumulation in Cu, corresponding to a minute imbalance of spins (moment) on each Cu atom

11.7 Application of XMCD to the Study of Transient Spin Effects

581

the theory side, the transient spin effects in Cu are so small that they presently cannot be reliably calculated by first-principles theory. To avoid these problems, recent theoretical work instead investigated the Co/Pd bilayer system which yields larger effects due to the higher spin polarizability of Pd [77].

11.7.2 Spin-Orbit Induced Spin Currents in Pt, Injected into Co Our second example of high sensitivity XMCD studies of spin current-induced effects deals with the spin Hall effect, predicted by Russian physicists Mikhail Dyakonov and Vladimir Perel in 1971 [78].11 These scientists also introduced the notion of a pure spin current that is decoupled from a charge current. For the Co/Cu system discussed above, the spin current is created through scattering of electrons on magnetic atoms. In contrast, a spin current may also be created through the spin-orbit interaction of flowing electrons with heavy atoms. This socalled spin Hall effect does not require the atoms to be magnetic, and in contrast to the conventional Hall effect, no magnetic field is required. Since the s–o interaction increases with atomic size, one uses materials consisting of heavy atoms such as Pt [80]. The spin dependent deflection of electrons flowing through a metallic Pt wire of rectangular cross section is illustrated in Fig. 11.19a. It leads to spin accumulation at the wire surface with directions indicated by red arrows s. The spin accumulation at the top Pt surface has been measured by Stamm et al. [81] by its effect on a ferromagnetic Co layer added on top, shown in blue. The magnetization direction of Co was aligned in the direction of electron flow, x, by an external magnetic field Hext , as shown in the figure. The spin accumulation at the Pt surface leads to spin diffusion into Co, and owing to the perpendicular orientation of the injected spins s and the Co moments m, a torque is exerted on the direction of the Co moments. They rotate slightly toward the surface normal z from their original x direction, as indicated. The slightly increased component of m along z is detected by XMCD by use of the shown normal incidence “forbidden geometry” which is almost exclusively sensitive to the m z component. The change of the spin torque-induced component of the Co moments m z with current through Pt is shown in Fig. 11.19b. By use of 50 kHz modulation of the current through Pt, it was possible to measure the tiny XMCD effect of order of 10−5 with a remarkable signal to noise ratio. Comparison of the current densities and dichroism effects with those in Fig. 11.18c reveals an even higher sensitivity. The measured XMCD signals were converted into the transient spin and orbital Co moments plotted as a function of current density in Fig. 11.19c, assuming that for metallic Co the spin density Tz  can be neglected. Both transiently induced Co moments were found to scale linearly with current density. 11

The term “spin Hall effect” was introduced by J. E. Hirsch in 1999 who repredicted the effect [79].

582

x-rays z

(a)

10

6nm thick

m

s

n flow 20 electro 0n spin m m rando dulation o m 50kHz 5

x

2.5nm thick

Pt

1x10

O

y

Co

Out-of-plane AC XMCD signal

Fig. 11.19 a Experimental configuration used by Stamm et al. [81] to measure the spin Hall effect with XMCD. b Co XMCD difference signal along z, measured by 50 kHz modulation of the current through Pt with current density. c Extracted changes of the spin and orbital Co moment as a function of current density

11 Quantum Theory of X-Ray Dichroism

Hext

spin Hall accumulation

-

(b)

0

-1x10

current density

5 -

6

2

6

2

6

2

0.79x10 A/cm

-2x10

2.35x10 A/cm

5 -

3.78x10 A/cm

m (

B)

775

780

6x10 4x10 2x10

4

(c) -

4

-

4

-

785 790 795 800 Photon energy (eV)

805

810

spin moment

orbital moment 0 0

6

6

6

1x10 2x10 3x10 2 Current density (A/cm )

4x10

6

11.8 X-Ray Magnetic Linear Dichroism—XMLD 11.8.1 Introduction The XMLD effect was first predicted and observed in 1985/86 by Thole and van der Laan et al. [82, 83]. In contrast to natural linear dichroism or XNLD, which arises from an anisotropy of the valence charge, magnetic linear dichroism or XMCD only exists in the presence of spin alignment in a sample. It is defined as the difference in cross section or intensity of two spectra measured with the E-vector oriented parallel and perpendicular to the spin alignment axis. Since for linearly polarized light the E oscillates in time along an axis, the XMLD effect only depends on the relative orientations of the E axis and the magnetic axis of the sample. This leads to the

11.8 X-Ray Magnetic Linear Dichroism—XMLD

583

dependence of XMLD on the square of the magnetization. In a simple picture the effect arises from a uniaxial distortion of the charge density relative to the spin axis as schematically shown in Fig. 11.20. The XMLD effect arises from a distortion of the atomic charge by the spinorbit interaction relative to a spin alignment axis of the sample. The XMLD signal is defined as the difference of the XAS spectra recorded for parallel and perpendicular alignment of E relative to the spin axis. Spin alignment may be created in paramagnets by application of an external magnetic field or is naturally present in magnetic materials due to the intrinsic exchange field. In practice, magnetic fields produced in the laboratory are much weaker (less than 10 Tesla) than the intrinsic exchange field in ferromagnets or collinear antiferromagnets (of order 103 −104 Tesla). In practice, XMLD therefore has its most important applications in the study of antiferromagnets, which owing to their internal spin compensation do not exhibit an XMCD effect. The real power of XMLD emerges mostly in the presence of multiplet structure where pronounced changes of the relative intensity of different peaks may be observed, as first shown theoretically by van der Laan and Thole [84]. As discussed in Sect. 10.6, the essence of multiplet structure is the coupling of spin and orbital angular momenta which in the final state involves those of both core and valence electrons. Since XMLD results from the coupling of the spin and orbital degrees of freedom in the sample, the effect is stronger in the presence of multiplet coupling than in one-electron theory. In the latter, the s–o couplings in the core and valence shells enter separately, and in transition metal ions the valence shell s–o coupling is weak relative to the larger ligand field effects. This typically results in relatively weak XMLD effects when multiplet structure is absent.

Charge distribution in paramagnetic state

Charge distribution in aligned magnetic state

s Fig. 11.20 Illustration of the origin of the XMLD effect. A paramagnetic sample with a spherical charge distribution (top) will exhibit a uniaxially distorted charge distribution when the spins are preferentially aligned, as shown at the bottom. The distortion is due to the s–o interaction that couples the charge and spin degrees of freedom

584

11 Quantum Theory of X-Ray Dichroism

Materials may exhibit both a charge anisotropy due to the ligand field as well as a spin alignment direction, so that XNLD and XMLD effects may be simultaneously present. In theory one may distinguish two types of XNLD measurements [85] referred to as type I and type II XMLD which are illustrated in Fig. 11.21. Type I XMLD is measured with fixed x-ray polarization E as the difference between rotation of the spin axis parallel and perpendicular to E. It eliminates any XNLD effect since the E-vector is fixed. It may be used for paramagnetic and ferromagnetic samples where the spin direction can be rotated by a strong field. In antiferromagnets one can, in principle, also “flop” the magnetization axis, but this typically requires fields larger than available. Type I also contains useful information on the magnetocrystalline anisotropy [86–88]. Type II XMLD measurements are performed when it is difficult or impossible to rotate the spin alignment direction. One then rotates the polarization vector and measures the difference between its parallel and perpendicular alignment with the spin axis. The interpretation of the results may be complicated by the existence of competing XMLD and XNLD effects.

11.8.2 Theoretical Formulation in One-Electron Theory The XMLD absorption cross-section difference is obtained from two measurements with E-vector parallel and perpendicular to the spin or magnetic alignment axis 

⊥ (ω). ΔσXMLD = σXAS (ω) − σXAS

Type I XMLD: fix E, rotate H H

(11.69)

Type II XMLD : rotate E rel. to magnetic axis

Magnetic axis

E H

E

Fig. 11.21 The two methods to record XMLD spectra. Type I XMLD is a pure magnetic effect, measured with a fixed orientation of the linearly polarized E-vector relative to the sample whose spin orientation is rotated by a sufficiently strong magnetic field. Type II XMLD is typically measured when it is difficult to rotate the magnetic alignment direction in a sample as for antiferromagnets. One then rotates the E-vector instead. The measured cross sections are labeled  and ⊥ with the XMLD signal defined by (11.69)

11.8 X-Ray Magnetic Linear Dichroism—XMLD

585

Assuming magnetic alignment along the z-axis, the energy dependent XMLD difference intensity is given by the difference of the intensities measured for E  z and E ⊥ z. The cross sections are written in analogy to those for XNLD (11.3) and XMCD (11.37) by use of the polarization dependent dipole operators in Table 11.1. For E along z and x, we obtain ΔσXMLD = A

2  (1) d |C0 | p j , m j  − j,m j

2 1 (1) (1) d |C−1 −C+1 | p j , m j  , (11.70) 2

where A is defined in (11.2) and d is the wavefunction of the empty LF states to which transitions can occur.

11.8.3 XNLD Versus XMLD in Cu-Phthalocyanine We will see later that the largest XMLD effects arise in the presence of multiplet effects since they lead to a large s–o mixing of the core and valence states as discussed in Sect. 10.6. Multiplet effect, however, is typically handled by computer programs at the expense of simple physical pictures that underlie one-electron theory. For this reason we shall first discuss a case where one-electron theory applies. We again consider the pedagogical case of Cu-Phthalocyanine (Cu-Pc), whose XMCD sum rule analysis was presented in Sect. 11.6. Here our goal is to put into perspective the large natural linear dichroism effect in Cu-Pc caused by the LF anisotropy, discussed in Sect. 11.3.1, in terms of the much weaker magnetic linear dichroism effect which is solely due to the s–o interaction. Our comparison disentangles the two effects and reveals the fundamental role of the s–o coupling in XMLD. The L-edge spectrum of Cu-Pc shown in Fig. 11.3b reveals a pronounced polarization dependence associated with transitions to the partially empty orbital d˜x 2 −y 2 . The small residual intensity in the red spectrum in Fig. 11.3b is mostly due to the fact that E was aligned at a finite angle of 20◦ from the normal z [5]. However, there is also a smaller s–o contribution due to admixtures of the pure LF orbitals dx y , dx z , d yz to the dx 2 −y 2 orbital in the presence of an external magnetic field. While it is experimentally challenging to disentangle the two effects for Cu-Pc due to the difficulty of aligning the E-vector exactly along z, this can be done theoretically. In this “forbidden” geometry, the slight s–o rotation of the pure LF dx 2 −y 2 to the d˜x 2 −y 2 orbital will lead to a type I XMLD effect, revealed by the emergence of a small intensity. To illustrate this effect, we align E  z and differentiate between Hext  z and Hext  x. Depending on the polarity of the Hext field, the ground state of the valence shell will either be the spin-up or spin-down state, and we consider the spin-up state in (11.46) for Hext  +z given by ↑

↑

|Hext +z  = |dx 2 −y 2  +

  iξ ξ |dx↑y  + |dx↓z  − i|d yz↓  E0 −E2 2(E0 −E3 )

(11.71)

586

11 Quantum Theory of X-Ray Dichroism

Table 11.4 Squared angular transition matrix elements per spin between the 2 p core states and the five d valence orbitals (also see Fig. 11.2) Polarization Core state dx y dx z d yz dx 2 −y 2 d3z 2 −r 2 Ez Ey Ex

2p 2p 2p

0 3 15 3 15

3 15

0

3 15 3 15

3 15

0

0 3 15 3 15

4 15 1 15 1 15

and the spin-up state (11.59) for Hext  +x ↑ |Hext +x  =

↑ |dx 2 −y 2 

  iξ ξ ↓ ↓ ↑ − |d  + |d  − i|d yz  . E0 − E 2 x y 2(E0 −E3 ) x z

(11.72)

The spin-down states will yield the same results, as will be replacing the directions x and y. We now use these valence states to evaluate the matrix elements in (11.70). This is facilitated by use of Table 11.4 which lists the matrix elements previously shown in Fig. 11.2. We obtain the following result for Hext  +z :  2 2  ↑ 2 ξ (1) Ez: 0 + Hext +z |C0 | p j , m j  =  5 2(E0 −E3 ) j,m j XNLD   

(11.73)

XMLD

and Ex :

2 1  ↑ (1) (1) Hext +z |C−1 −C+1 | p j , m j  2 j,m j  2  2 1 1 1 ξ ξ = + + . 5 5 (E0 −E2 ) 5 2(E0 −E3 )     XNLD

(11.74)

XMLD

While the XNLD effect identified by the first underbracket is large, we find only a very small type II XMLD effect, since ε = 102 meV (see Table 9.1) and E0 −E2 = 2.0 eV is similar to E0 −E3 = 2.3 eV. Furthermore, it turns out that for the rotated magnetic field Hext  +x we obtain the same result as for Hext  +z, given for E  z (11.73) and E  x by (11.74). Hence there is no type I but only a small type II XMLD.

11.8.4 XMLD in Ferromagnetic Transition Metals Stronger XMLD effects may be observed when the magnetic alignment field increases as in ferromagnetic transition metals, and even in non-magnetic transition metals where magnetism is induced by hybridization with adjacent ferromagnets, as

11.8 X-Ray Magnetic Linear Dichroism—XMLD

587

observed by Schwickert et al. [89]. The origin of XMLD in ferromagnetic transition metals has been explained by use of band theory by Kuneš and Oppeneer [90]. Below we reduce their rigorous band model, which in essence is an independent electron model, to an atomic-like model illustrated in Fig. 11.22a which can be solved analytically in a one-electron formalism and contains the essence of the physics. We assume that the spin-down valence states are filled and describe the empty spin-up states by an equal weighting of all di orbitals, reflecting an effective averaging of the k-dependent states over the Brillouin zone. Since the s–o interaction in the 3d valence shell is small (∼ 50 meV) compared to the exchange interaction (1–2 eV), we neglect it. In a ferromagnet, the strong exchange field of order of 103 Tesla which only acts on the spin leads to a small exchange splitting of the two individual 2 p3/2 and 2 p1/2 core states. The resulting individual m j substates are no longer isotropic as illustrated

m +1/2

3d ~1eV

I-I

-1/2

= -3 4 -1 0

-2 2

mj

2p3/2 2p ~15eV 2p1/2

+3/2 +1/2 -1/2 -3/2

~0.2eV

-1/2 +1/2

Intensity and difference

(a)

Difference Intensity

X-ray Magnetic Linear Dichroism in a ferromagnet

(b) 4 3 2 1 0 -1 -2 -3

(c)

770

XMLD spectrum

L3

1 + 3 +1 2 2 2

I-I

L2

mj

- 12 + 12

Co metal

800 790 780 Photon energy (eV)

Fig. 11.22 a L-edge transitions from s–o split 2 p3/2 and 2 p1/2 core states (∼ 15 eV) to 3d valence states split by a strong exchange field (∼ 1 eV), with the magnetization direction m chosen so that the spin-down 3d states are filled and the spin-up states partially empty. The s–o splitting of the valence states has been neglected due to its small size (∼ 50 meV). The 2 p3/2 and 2 p1/2 core states are further split into m j substates (∼ 0.2 eV) by a strong exchange field (∼ 2×103 Tesla). Note the opposite order of m j states for p3/2 , l + s and p1/2 , l − s because of the opposite sign of s. The transition intensities are calculated for polarization E  z  m and E  x ⊥ m with the linear polarization operators in Table 11.1. The shown difference intensities multiplied (see Fig. A.8) by a factor of 90 correspond to the difference of the matrix elements in (11.70). b Schematic difference intensities according to (a). c Original data of XMLD for Co metal by Schwickert et al. [89]

588

11 Quantum Theory of X-Ray Dichroism

in Fig. 10.9. Owing to their separation of order 0.2 eV [90–92] and the associated difference in the transition matrix elements, the splitting changes the lineshape of the core to valence transitions as schematically shown in Fig. 11.22b. The transition intensity differences can be worked out by use of Fig. A.8. In practice, this effect is seen in the XMCD difference spectrum as a pronounced differential resonance lineshape at both the L3 and L2 edges. The same lineshape at the two edges is a consequence of the fact that for the p3/2 , (l + s) and p1/2 , (l − s) levels the signs of both s and m j are inverted. The cross section (11.70) of the largest peak in the XMLD (difference) spectrum is (4/90)A which compares to the value (2/9)A for the XMCD difference obtained from (11.26). In the ferromagnetic metals the XMLD effect is hence considerably smaller than the XMCD effect. In addition to x-ray absorption, XMLD spectra can also be measured by analyzing the polarization of the x-rays transmitted through the sample. Such magneto-optical polarization spectroscopy can be performed in several different geometries [93] and is complementary to x-ray absorption spectroscopy.

11.8.5 Enhanced XMLD Through Multiplet Effects In our discussion above we have used a simple one-electron model and seen that XMLD is typically quite small in this case. As mentioned earlier, XMLD effects may be significantly larger in the presence of multiplet structure. Such structure is typically present in the x-ray absorption spectra of transition metal oxides [3, 94], which are often antiferromagnetic. One of the most beautiful applications of XMLD is the imaging of antiferromagnetic (AFM) domains, first accomplished around 2000 [95–99] (see Fig. 8.4c). Following spectroscopy studies of cubic NiO by Alders et al. [100], early AFM domain images [98, 99] were interpreted by linking relative intensity changes of multiplet peaks to the direction of the AFM axis. More recently, it has been shown by detailed experimental and theoretical studies on NiO by van der Laan, Arenholz, and co-workers that the situation is considerably more complicated by the superposition of two fundamental spectra with distinctly different spectral intensities [101–104]. This led to a reinterpretation of the AFM axis direction by 90◦ , as reviewed by van der Laan [64] and shown in Fig. 1.21d. The revised orientation of the AFM axis in NiO is in agreement with that found by neutron scattering [105] and the perpendicular exchange coupling between AFM NiO and ferromagnetic Co theoretically predicted by Koon [106] and Schulthess and Butler [107]. Another family of antiferromagnets are Fe-based oxides. The linear polarization dependence of XAS spectra of Fe2 O3 (hematite) was first investigated by Kuiper et al. [108] and revised more recently in both experiment and theory by Arenholz et al. [109]. As for the NiO case it was found that the XMLD spectra consist of a superposition of two fundamental multiplet spectra. Similar multiplet structure as Fe2 O3 is found in the much studied rare earth (R) orthoferrites, RFeO3 . As an example we show in Fig. 11.23 the strong XMLD effect

11.8 X-Ray Magnetic Linear Dichroism—XMLD

589

LaFeO3 SrTiO 3 (110) [110] z

[110]

25 Fe

Normalized Intensity

L-edge

y [001]

x

E||x E||y E||z

4

20

3

15

2

10

1 721

723

725

5 0

710

715 720 Photon Energy (eV)

725

Fig. 11.23 Polarization dependent Fe L2,3 XMLD spectra of LaFeO3 /SrTiO3 (110) [110]. For the three shown spectra the electric field vector of the linearly polarized x-rays was oriented along the three indicated crystallographic axes

of the multiplet peaks in LaFeO3 grown on cubic SrTiO3 (110), taken from the work of Lüning et al. [110]. Owing to the local cubic coordination of the Fe by O atoms, the measured polarization dependence is of magnetic origin, also shown by its disappearance above the reduced Néel when doped with Sr. The LaFeO3 /SrTiO3 (110) system was also used to demonstrate AFM domain imaging by Nolting and Scholl and collaborators [96, 97, 111]. Later studies revealed the sensitivity to the AFM axis orientation to strain [112] and the influence of different growth substrates (e.g. MgO) on domain size and exchange bias when coupled to ferromagnetic Co [113]. In summary, it appears that the enhanced XMLD effects in oxides arising from multiplet effects come at the price of spectral complexity, which typically requires a detailed comparison of experiment with theory.

590

11 Quantum Theory of X-Ray Dichroism

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Chapter 12

Quantum Theory of X-Ray Emission and Thomson Scattering

12.1 Introduction and Overview Following our discussion of XAS and its polarization dependence in the last two chapters we here continue with the discussion of two other important first order processes, namely x-ray emission spectroscopy (XES) and Thomson scattering, outlined in Sect. 9.5 and illustrated schematically in Fig. 9.4b, c. X-ray emission is the inverse of the x-ray absorption process, and both can be described within KHD theory by transitions between two states. The interaction Hamiltonian for the two processes is the same, given by H int = −e r·E in (9.22). In XES, one starts with an initial state that is an excited state with a core hole in an inner shell and considers its decay to another excited electronic state, where the core hole is now in an outer electronic shell, typically the valence shell. Our treatment of XES covers the quantum mechanical derivation of the emission rate as well as the tricky problem of the emitted linewidth. We will show that, in general, the XES linewidth contains additive contributions from the initial and final core hole states. Particularly remarkable is that the XES linewidth is not solely determined by the dipolar transition width that describes radiative decays, but that it always has a “ghost contribution” from the Auger decay channel. We finish our discussion of XES with examples of fundamental XES experiments of atomic and molecular gases and solids. We then discuss the quantitative separation of the radiative XES and non-radiative Auger channels, expressed by the so-called fluorescence yield. It is a measure of the total XES probability averaged over polarization, emission directions, and energies and is defined as the dimensionless ratio of the total x-ray emission probability relative to the total x-ray plus Auger emission probability. In the process, we discuss how our quantum mechanical XES results compare to tabulated semi-empirical values. The last part of the present chapter is devoted to the quantum derivation of atomic Thomson scattering. Like XAS and XES, it is also described as a first order process

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_12

595

596

12 Quantum Theory of X-Ray Emission and Thomson Scattering

within KHD theory, but the interaction Hamiltonian is given by e2 E 2 /(2m e ω2 ) in (9.22). The quantum formalism is shown to give the same results as the semi-classical formalism for elastic Thomson scattering, previously discussed in Sect. 6.2.4. While elastic Thomson scattering, which underlies Bragg diffraction, has historically had larger impact in x-ray science than inelastic Thomson scattering, we shall also discuss the latter. It is based on detecting the change in energy, direction, and polarization of the scattered photons. The lost energy reflects fundamental excitations in a sample, ranging from phonons and magnons to more energetic excitations of valence and even core electrons. As a particularly interesting application we discuss inelastic Thomson scattering in the form of so-called x-ray Raman scattering (XRS) also called non-resonant inelastic x-ray scattering (NIXS), which are shown to give the same results as XAS spectroscopy in the limit of small momentum transfer.

12.2 Quantum Formulation of X-Ray Emission Spectroscopy (XES) 12.2.1 XES History and Terminology We start by defining what we mean by “x-ray emission spectroscopy”, or XES for short. This is required because modern spectroscopy experiments are typically performed differently than early x-ray emission experiments. Following early studies by Barkla, Moseley, and Manne Siegbahn in the 1910–1920 period, high-resolution x-ray emission experiments were pioneered by the group of Kai Siegbahn at Uppsala University, Sweden, in the 1970s. In these studies, core holes were conveniently produced by electron impact ionization. With the advent of synchrotron radiation, x-ray emission studies increasingly involved better defined core excitations by x-rays. It became then important to distinguish different ways of preparing the core hole state since they required different quantum treatments. In this book we define XES as the study of the emission process that follows after core electron ionization. The excitation process may involve electron or photon ionization, and the excitation process does not enter into the XES description. This case is distinguished from resonant excitations of core electrons into unfilled localized valence states, referred to as resonant inelastic x-ray scattering or RIXS. In RIXS, the excitation step is specifically included since the excited core electron influences the decay, as discussed in Chap 13. The key theoretical difference is that XES may be described by first order KHD theory, while RIXS requires a second order description.

12.2 Quantum Formulation of X-Ray Emission Spectroscopy (XES)

597

We define the term “x-ray emission spectroscopy”, or XES, to describe the case where the excitation process does not enter into the description of the atomic core hole decay process. The decay process starts from an initial state that contains a core hole and may have photons present in a polarization and wavevector mode pk. The final XES state consists of one more photon and an electronic hole state in an outer atomic shell. As in the formulation of XAS in Sect. 10.2, the quantum mechanical description of XES may be accomplished through the first order term in the KHD formula (9.21) and utilizes the interaction Hamiltonian written as Hint = −e r · E in (9.22). The KHD transition rate is given by (10.1) or Wi f =

2π |  f |e r·E|i|2 ρ(E f ) δ(E f − Ei ). 

(12.1)

In contrast to x-ray absorption, the general field operator given by E = E+ + E− is now given by the photon creation part E− ∝ a†pk . The initial XES state |i in (12.1) is a product state of the upper electronic state |b and a photon number state |npk  containing npk photons in the mode pk. The decay may occur without the presence of any photons. The initial photon state is then the zero-point quantum vacuum |npk = 0. The final XES state is the product of the lower electronic state |a and the photon state |npk + 1 since a photon has been generated in the decay.

12.2.2 The Photon Part of the Transition Matrix Element The photon part of the transition matrix element in (12.1) is evaluated in analogy to the XAS case (10.2) as   n pk + 1|E− |n pk 2 = (n pk + 1)ω , 20 V pk

(12.2)

where V pk is given by (3.106). The factor n pk + 1 is the sum of the spontaneous and stimulated contributions to emission. Before we further evaluate (12.1) into separate photon and electronic state components, let us take a closer look at the fundamental and well-known term 1 + n pk . 12.2.2.1

Spontaneous and Stimulated Decays: Zero-Point Field

The KHD theory conveniently utilizes the concept of labeling photons by their pk modes reflecting the quantization or division of the EM field into “coherent” fractions. The factor 1 + n pk means that emission of a photon may be induced two ways,

598

12 Quantum Theory of X-Ray Emission and Thomson Scattering

either spontaneously by the zero-point (ZP) field, with a relative probability 1, or by stimulation of the decay by n pk real photons, with a probability n pk . The distinction between stimulated and spontaneous emission was introduced by Einstein in 1917 [1, 2] in his derivation of Planck’s blackbody radiation law [3, 4]. There is an important difference between the nature of the photons that are created in spontaneous and stimulated emission. In spontaneous emission, the photon may be emitted into any direction, owing to the random nature of the ZP field from which it is created. In contrast, stimulated emission is a cloning process that results in the emission of two indistinguishable photons which have been referred to as “cloned biphotons” [5, 6]. In spontaneous emission, photons are emitted into random directions and with random polarizations. The quantum mechanical average corresponds to the classical emission of a spherical wave. In stimulated emission, the created photons represent clones of the incident ones and hence are emitted into the same mode, corresponding to the classical emission of a plane wave. The factor 1+n pk reveals that the stimulated decay probability induced by a single real photon, n pk = 1, is the same as the spontaneous one induced by virtual photons of the ZP field which contains one half photon per mode pk or one photons per k-mode. Hence the ZP field contains one photon per k-mode. We can express the quantized ZP field in analogy to (3.62) as k 2 | = |E ZP

ω . 20 V pk

(12.3)

The incident field containing n k photons in the same polarization mode is similarly given by |E 0k |2 = n k

ω . 20 V pk

(12.4)

Therefore n k is simply a measure of the relative field strength of the incident to zero-point field in the mode k according to nk =

|E 0k |2 . k 2 |E ZP |

(12.5)

While it appears that it should be easy to overcome the ZP field and change spontaneous to stimulated emission by use of modern synchrotron sources, this is not so. The reason is that it takes an XFEL to increase the photon degeneracy parameter, defined in Sect. 4.2.7 as the number of photons in the same mode, i.e. n pk , to values larger than 1, as illustrated in Fig. 4.5.

12.2 Quantum Formulation of X-Ray Emission Spectroscopy (XES)

599

12.2.3 XES Decay Time and Linewidth Next we consider the factor δ(E f − Ei ) ρ(E f ) in (12.1). The δ-function assures energy conservation in the emission process between all photon and electronic degrees of freedom. The density of states factor ρ(E f ) in (12.1) reflects the total density per unit energy of the combined photon and electronic systems after emission. We can, however, take advantage of the fact that energy is also conserved between the electronic and photon systems, so that we can separately consider them as we have done for the transition matrix element in Sect. 12.2.2. For our discussion we adopt the notations used for the XAS case in Fig. 10.3. The electronic system is initially in the excited core hole state |b. It contains a certain stored energy Eb . Since the core hole state can decay, its energy has an intrinsic uncertainty width. We first assume for simplicity that the electronic final state |a is the electronic ground state. Since it does not decay, it has an infinitely long lifetime, and there is no uncertainty in its energy Ea . We consider the decay process illustrated by the electronic diagram shown in Fig. 12.1a. In the figure we have indicated an uncertainty b (dimension [energy]) in the initial state energy Eb . In a semi-classical model, the energy uncertainty corresponds to a decay rate b / (dimension [1/time]) of state |b. At this point, b is a parameter and we do consider its deeper meaning. We keep in mind, however, that we later need to clarify two issues associated with our assumed model. First, we need to consider that the final state |a is not the ground state but may be another excited electronic ground state of finite lifetime. Second, we need to address that core hole decays actually involve competing radiative and Auger decay channels, as previously discussed in Sect. 10.4.1 and illustrated in Fig. 10.3. The wavefunction or probability amplitude of our two-level system at t ≥ 0 is given by  b i ωba t −b t/(2) e . (12.6) e (t) =  The system oscillates with the resonance frequency ωba , where Eba = ωba is the energy separation of the excited electronic state |b from the ground state |a. At t > 0 the amplitude of the wavefunction decreases exponentially with time and integration of the probability |(t)|2 over time shows that the sum of upper and lower state populations is conserved, as required, ∞ |(t)|2 dt = 0

b 

∞

e−b t/ dt = 1.

(12.7)

0

The temporal wavefunction (t) is given in the frequency domain by the Fourier transform

12 Quantum Theory of X-Ray Emission and Thomson Scattering

Fig. 12.1 a Decay diagram of an upper electronic state |b with an energy uncertainty of b to a lower state |a. b Lorentzian shape of the emitted radiation of FWHM b . c Decrease of the emitted probability amplitude (12.6) in the form sin(ωba t) exp[−b t/(2)] for b = 1 eV and ωba = 100 eV with time t in fs. The coherence time / b over which the upper population decays to 1/e is shown in red. d Change of the upper and lower state populations with time

(b) Emitted spectral line

(a) Decay diagram 1.0

b b

h

ba

Probability

600

0.5 b

0.0 -2 -1 0 Energy difference

a coh=

1

ba

2 (eV)

-h

h/ b = 0.66 fs

1.0

Probability amplitude

=1eV

(c) Probability amplitude

0.5 0 -0.5 -1.0 1.0

upper state b

Population

0.8

(d) State populations

0.6 0.4 0.2 0.0 lower state a 0.0 0.5

 (ω) =

b 

1.5

2.0

b  . i(Eba − ω) + b /2

(12.8)

√

∞ e 0

1.0 Time (fs)

i ωba t

e

−b t/(2) −i ωt

e

dt =

The associated probability distribution has a Lorentzian shape of FWHM b given by (b /2)2 2h . (12.9) | (ω)|2 = π b (Eba − ω)2 + (b /2)2 Now we invoke energy conservation between the electronic and photon systems. Since the electronic system ends up in the zero-energy ground state, the expression (12.9) reflects the probability of the emitted photon energy distribution. It is illustrated

12.2 Quantum Formulation of X-Ray Emission Spectroscopy (XES)

601

in normalized form in Fig. 12.1b. The final state quantity ρ(E f )δ(E f − Ei ) in (12.1) can then be expressed in the form ρ(E f )δ(E f − Ei ) =

(b /2)2 | (ω)|2 2 . = h π b (Eba − ω)2 + (b /2)2

(12.10)

It has the dimension [1/energy] and fulfills the normalization ∞ ρ(E f ) δ(E f − Ei ) d(ω) = 1

(12.11)

−∞

corresponding to one emitted photon. The decay process may be pictured as illustrated in Fig. 12.1c, d with the probability amplitude oscillating between the excited and ground state during the emission process with concomitant change of the excited and ground state populations. In the following two sections we extend our simple model to the cases that the final state is another excited electronic state and that there is also an Auger decay channel.

12.2.4 Decays to Excited Final States We now consider that the lower electronic state is another excited state of finite lifetime that may decay further, which is the situation encountered in practice. This problem was first addressed from a quantum theoretical point of view in 1930 by Weisskopf and Wigner [7, 8]. We may treat the case of a finite lifetime of the lower state |a by introducing an associated decay width a as illustrated in Fig. 12.2. It illustrates that we now have energy uncertainties expressed by Lorentzian density distributions in both upper and lower states, leading to an increased energy uncertainty (width) of the emitted photons. From Fig. 10.4 we know that the decay width  of electronic states increases with their energy E0 above the ground state with an approximate correlation /E0 ≈ 10−3 . For a core hole energy of Eb ∼ 1 keV in Fig. 12.2 we thus have b ∼ 1 eV and for a typical final state energy Ea ∼ 1−10 eV we expect a ∼ 1−10 meV. Hence in XES, the core hole decay width will typically dominate. In other words, core holes will decay much faster (femtoseconds) than valence holes (picoseconds). Let us nevertheless consider the general case following Weisskopf and Wigner [7, 8]. By inclusion of energy conservation, indicated by the length of the red arrow in Fig. 12.2, we need to integrate over all energies Ex and obtain the following expression instead of (12.10)

602

12 Quantum Theory of X-Ray Emission and Thomson Scattering b

|b

x

b

h

ba a

|a

a

x- h

Fig. 12.2 Atomic level diagram for decay from an upper electronic state |b with an associated Lorentzian FWHM b to a lower state |a of Lorentzian FWHM a . We have indicated the energy uncertainties by Lorentzian density distributions in both upper and lower states. The red transition arrow conserves energy according to the δ-function in (12.1)

| (ω)|2 2 2 = h π a π b

∞ −∞

(b /2)2 (a /2)2 dEx . 2 2 (Ex − Eb ) + (b /2) (Ex − ω − Ea )2 + (a /2)2 (12.12)

This is seen to be a convolution of two Lorentzians which is another Lorentzian and with the notation Eba = Eb − Ea we obtain instead of (12.10) ρ(E f )δ(E f − Ei ) =

2 (a + b )2 /4 . (12.13) π(a +b ) (Eba − ω)2 + (a + b )2 /4

Our new linewidth expression contains the sum of the upper and lower state decay widths. It reduces to our previous expression (12.10) in the limit a = 0 where |a is the stable ground state. In retrospect, our previous result for a ground state of infinite lifetime is seen to correspond to a convolution of the form (12.12) where the final state Lorentzian has the form of a δ-function.

12.2.5 Auger Contribution to the XES Linewidth We now address a final issue in the derivation of the XES linewidth, namely that core hole decays involve both radiative and Auger channels, as previously introduced in an ad hoc fashion in Fig. 10.3. The decay problem is illustrated in detail in Fig. 12.3 and requires consideration of both radiative x-ray and non-radiative Auger decay channels of the initial core hole state |b and final state |a. While radiative transitions are driven by the dipole operator, e r · E, Auger transitions involve the Coulomb operator, e2 /r . The proper quantum treatment of the XES linewidth requires taking account of both photon and electron emission channels by a second order treatment. Our first order description of XES in the KHD formalism worked because it tacitly used the

12.2 Quantum Formulation of X-Ray Emission Spectroscopy (XES)

|a

X-ray emission linewidth | b b+

=

b=

|b radiative decays

603

a X A b + b

Auger decays

h

~

X

|a

a= a

h

|f’

’

+

|f

A

a

’

~

|f’

Fig. 12.3 Illustration of the radiative and Auger decay channels involved in the derivation of the x-ray emission linewidth of a dipole transition from an initial core hole state |b to a final electronic state |a (red) that is not the ground state. A given excited electronic state may decay either by emission of a photon (wiggly lines) of energy ω or Auger electron (straight arrows) of energy denoted E. The measured x-ray emission width for the transition b → a shown in red is  = b + a and contains additive dipolar and Auger contributions for both the initial and final states of the transition as indicated and given by (12.14)

second order linewidth result which is  =  X +  A . In the literature, the proper linewidth derivation is typically circumvented by adopting this result in an ad hoc fashion. We also will not go through the detailed second order linewidth derivation and just state the key results.1 It turns out that one may associate both an energy En and intrinsic energy width n with any given electronic state |n where each n is again the sum of a radiative and Auger contribution, n = nX + nA . In Fig. 12.3, the total decay width from |b to |a is then given by the following remarkable result. The XES linewidth of an electronic transition |b → |a is given by a Lorentzian of FWHM (12.14)  = a + b = aX + bX + aA + bA .       dipole width

Auger width

Although only the x-ray decay channel is measured, Auger decays contribute an inseparable “ghost” linewidth contribution. Similarly, the Auger linewidth contains a “ghost” radiative contribution. The fascinating ghost contributions to the measured linewidth in the alternative decay channels arise from the quantum mechanical link established in the birth of 1

I would like to thank Faris Gel’mukhanov for sending me his full quantum derivation of the x-ray emission linewidth.

604

12 Quantum Theory of X-Ray Emission and Thomson Scattering

the decay process. Such a quantum link through birth also occurs in spontaneous parametric down conversion, where the two born photons are entangled. Each of the so-created photons also carries a ghost contribution from the other photon, exploited today in so-called ghost imaging [9, 10]. The proof of the existence of the entangled ghost phenomenon was honored by the 2022 Nobel Prize in Physics awarded to Alain Aspect, John F. Clauser, and Anton Zeilinger.

12.2.6 Putting It All Together: The X-Ray Emission Rate and Width We are now in a position to complete the evaluation of the KHD emission rate (12.1). The final state density is given by (12.10), with  given by our final result (12.14). In the derivation of the Lorentzian form we implicitly assumed energy conservation, so that the δ-function in (12.1) has already been incorporated. The transition matrix element is factored into a photon part, given by (12.2), and the electronic part may be written in the form (10.40) of the absorption case. Putting it all together we obtain as follows. In the dipole approximation, the spontaneous plus stimulated XES rate is given by p (1+n pk )c λ2 ba (/2)2 XES = , (12.15) Wb→a V pk π  (ω − Eba )2 + (/2)2       Flux, pk

p

σXES

where  = a + b is given by (12.14). The generalized XES cross section including stimulated emission may be defined as p

σXES =

p

λ2 ba (/2)2 1 + n pk , 2 2 π  (ω − Eba ) + (/2)

(12.16)

where the radiative dipolar width is given by p

ba =

8π 2 ω 2  λ  

|a|r ·  p |b|2 . αf     coupling atomic part photon includes spin part

(12.17)

The form of the dipolar matrix element reveals the photon and electronic contributions coupled by the fine structure constant, just like the corresponding XAS expression (10.40).

12.2 Quantum Formulation of X-Ray Emission Spectroscopy (XES)

605

abs There is great similarity between the absorption rate Wa→b given by (10.38) and em the emission rate Wb→a (12.15). However, for XAS, the lower state |a is the ground state of infinite lifetime and we have  = b = bX + bA . In XES, the total linewidth  is instead given by the full expression (12.14) or  = a + b . As discussed in Sect. 12.2.4, we typically have b  a , so we can summarize as follows.

The XES linewidth is typically dominated by that of the initial core hole state, b , which is larger than that of the final valence hole state, a . This is a consequence of the approximate scaling of the decay linewidth of a level with its binding energy (see Fig. 10.4). The fundamental XES linewidth  that determines the decay rate and emission cross section in (12.16) is only observed when the emission spectrometer has an instrumental Gaussian resolution function of width much less than . If this is not the case, the theoretical Lorentzian width needs to be convoluted with the Gaussian instrumental width, resulting in a broadened Voigt lineshape [11] (see Appendix A.2.3).

12.2.7 Atomic Decay Time: Core Hole-Clock As discussed above, we typically have  b  a in XES since the decay energy of the initial core hole state |b is larger than that of the final state |a. The core hole width then represents an atom-specific lifetime τ according to the relation, τ =

 

(12.18)

The decay time τ is called the natural lifetime and constitutes an atom-specific reference or “core hole clock” time.2 As illustrated in Fig. 12.1d, it represents the time interval during which the probability (or population) of the excited core hole state decays to 1/e of its initial value and as shown in Fig. 12.1c may be associated with the coherence time of the emitted radiation. By use of (12.18) we can convert the atomic  values in Table 10.2 (also see Fig. 10.4) into characteristic atomic decay times which are shown in Fig. 12.4 for 1s (K-shell) and 2 p3/2 (L3 shell) core holes as a function of atomic number Z . For the shown cases the dominant mechanism is Auger decay so that in (12.14) we have  bA  bX . In the figure we have also shown in red on the right scale the low probability that the decay occurs by radiative x-ray emission. This probability is called the fluorescent yield as discussed in Sect. 12.4. 2

The name emerged from studies of electronic relation processes of core holes in adsorbates on surfaces [12, 13].

7

C N O F Ne Na Mg Al Si P S Cl Ar

K-edge

6

10

-1

10

-2

5 4 3

Total decay lifetime

2 1

6 4

8

Ca Sc Ti

10

12

14

16

18

V Cr Mn Fe Co Ni Cu Zn

L3-edge

3.5

10

-2

10

-3

3

X-Ray emission probability

Fig. 12.4 K- and L3 -shell core hole clock life times (black, left scale) calculated by use of (12.18) from the  values in Table 10.2. The probability that the decay occurs by radiative x-ray emission is shown in red (right scale). The red curve is also the fluorescent yield as discussed in Sect. 12.4 and shown in Fig. 12.10

12 Quantum Theory of X-Ray Emission and Thomson Scattering

(fs)

606

2.5 2 1.5 1

20

22

24 26 28 Atomic number Z

30

A beautiful experimental verification of (12.18) is shown in Fig. 12.5, measured by R. Röhlsberger using Mössbauer spectroscopy [14]. In the shown example, an excited nuclear state decays to a long-lived lower state by emission of a γ -ray photon or emission of a conversion electron. In complete analogy to an electronic state decay, the total lifetime of the excited state is determined by the sum of the radiative (photon emission) and non-radiative (electron emission) channels. Figure 12.5a shows the 57 Fe Mössbauer spectrum measured in the energy domain by the conventional transmission method. The time domain measurement, shown in (b) for the same sample, was carried out by exciting the 14.4125 keV Mössbauer transition by a narrowband synchrotron radiation pulse of about 100 ps, and then measuring the time dependence of the decaying intensity. The decay linewidth in the energy domain is readily obtained from the conventional single-line 57 Fe Mössbauer spectrum of stainless steel shown in Fig. 12.5a. The spectrum is almost a perfect Lorentzian of FWHM  = 18.9 neV. The time domain measurement, shown in (b) for the same sample, yields a 1/e decay time of

12.3 Fundamental X-Ray Emission Experiments 57

607

Fe Mossbauer spectra of stainless steel in energy and time domain 4

10

1

( b) 3

10

0.9

=18.9 neV

0.85

Counts

Relative Transmission

(a) 0.95

= 34.8 ns 2

10

0.8 0.75 0.7 -100

-50

0 Energy (neV)

50

100

10

0

40 80 120 Time after excitation (ns)

160

Fig. 12.5 a Mössbauer spectrum for stainless steel recorded in transmission for a 9.5-μm-thick stainless steel foil in the energy domain, yielding the FWHM value . b Measurement of the time dependent excited state decay after excitation with narrowband synchrotron radiation pulses, with an exponential fit (red), yielding the 1/e decay time constant τ . The two quantities are related according to (12.18) with τ = τ . In both cases the measured linewidth is about four times the natural linewidth of 4.7 neV. Courtesy of R. Röhlsberger, DESY

τ = τ = 34.8 ns. By use of the value  = 658.2 neV ns, the measured temporal and energetic widths can be directly converted into each other by (12.18).3

12.3 Fundamental X-Ray Emission Experiments 12.3.1 K-shell Emission in Ne Emission spectra of atoms are particularly simple since they are not influenced by vibrational fine structure which is present in molecules. Furthermore, the decay width of the 1s core state can be directly determined from the measured spectrum since the decay typically ends in an electronic state with a long lifetime, so that to a very good approximation we have  Xf =  Af = 0 in (12.14). Finally, the decay of the 1s core hole proceeds predominantly (∼98%) through the Auger process so that the emission linewidth is  iA , providing a beautiful demonstration that it is almost entirely due to the “ghost” linewidth contribution of the Auger channel. As a pedagogical case we consider the Kα1,2 x-ray emission spectrum of Ne atoms, as illustrated in Fig. 12.6. The Ne emission spectrum is also of interest because it has been utilized to demonstrate its amplification through stimulated emission [15] as previously shown in Fig. 1.29.

Note that the natural lifetime τ over which the excited state population decays to 1/e of its original value differs from the often quoted (shorter) half-life τ1/2 = ln2 τ of radioactive decays.

3

608

12 Quantum Theory of X-Ray Emission and Thomson Scattering

Neon K-shell x-ray emission (a) Energy level diagram ionization

Binding energy (eV)

vacuum level

0

(b) Decay configurations 1s1 2s 2

1s

~ 270 meV

1s

~ 2.4 fs

2p

~0

2p

very long

2p6

h dipole transition =848.6eV

2p3/2 2p1/2

- 21.6 - 21.7

1s 2 2s 2

2s

- 48.5

2p5

1s 2

h

ground state

2s 2

1s core hole

2p6

Ne K

1,2

Intensity

(c) Spectrum

- 870.2

850 845 860 855 Emitted photon energy (eV)

Fig. 12.6 Illustration of K-shell emission in Ne atoms. a Ne energy level diagram and levels involved in Kα1,2 emission. b Diagram of the electron/hole configurations involved in the decay of the 1s core hole. Electrons are indicated by filled black circles and the hole by an open red circle. The upper configuration is the initial state in the x-ray emission process. It decays exponentially via a dipole transition to the middle configuration which is metastable with a long lifetime as discussed in the text. c Kα1,2 emission spectrum recorded after electron impact ionization with an experimental resolution of 150 meV by Ågren et al. [16]. The 2 p3/2 and 2 p1/2 spin-orbit components separated by 100 meV are not resolved. The emitted K α1,2 doublet of ω 848.6 eV has two unresolved components, each of linewidth 1s = 0.27 ± 0.01 eV

12.3 Fundamental X-Ray Emission Experiments

609

Figure 12.6a illustrates the origin of the Ne K-shell x-ray emission line (Kα1,2 ) of ω 848.6 eV. It corresponds to the decay of a Ne 1s core hole configuration to the 2 p hole configuration, with spin-orbit components 2 p1/2 and 2 p3/2 separated by 100 meV resolved by high-resolution photoemisson (see Fig. 12.4 in [17]). The core decay in a configuration picture is shown in Fig. 12.6b. It illustrates that the final configuration containing a 2 p hole is long lived [18]. This is apparent from the extremely small 2 p1/2 and 2 p3/2 photoemission linewidths 2 p 5 meV shown in Fig. 12.2 of [17]. Hence we have 2 p 0, and the x-ray emission width is that of the initial state core hole  = 1s . The Kα1,2 emission spectrum was measured after electron impact ionization by Ågren et al. [16], as shown in Fig. 12.6c. In this 1978 experiment the instrumental (Gaussian) resolution was a remarkable 150 meV. A theoretical fit of the emission spectrum that took into account the unresolved infinitely sharp 2 p1/2 and 2 p3/2 spinorbit components with a 100 meV separation yielded a value of 1s = 270 ± 10 meV for the natural K-shell linewidth. This value compares to the values 248–297 meV derived from the width of the Rydberg XAS resonances in Fig. 10.10. The related case of K-shell (binding energy 3206 eV) excitations in Kr atoms has also been studied. Again good agreement is found for the natural K-shell lifetime width  determined by Rydberg absorption studies [19] ( = 680 ± 30 meV) and the 1s x-ray photoemission linewidth ( = 675 ± 20 meV) [20].

12.3.2 K-Shell Emission in N2 As a second example we discuss the K-shell emission in the N2 molecule, illustrated in Fig. 12.7. Like the Ne spectrum in Fig. 12.6 it was measured after electron beam ionization by the pioneering Uppsala group [21]. The molecular orbitals and their relative energies determined by photoemission spectroscopy [25] are shown in Fig. 12.7a. In particular, we also show the small 1 σu − 1 σg core splitting of 97 ± 3 meV [26]. The N2 level structure is that previously shown in Fig. 10.11a. Here we consider the scenario where the molecule is ionized through excitation of a 1s electron in either the 1σg or 1σu state into the continuum of empty states above the ionization potential (IP) indicated in blue. The created core hole may be of either spin or parity. Figure 12.7b illustrates the initial and final state configurations of the XES process, arbitrarily assuming spin-down transitions. The two initial states with opposite parity 1s holes may decay via spin-conserving and parity allowed dipole transitions to the shown three final state configurations with holes in the 2σu , 1πu , and 3σg orbitals. The latter have a long lifetime so that the minimum XES linewidth is determined by that of the 1s core hole width 1s , alone. The width has the value 1s 120 meV, determined from vibrationally resolved XAS [22] and PES spectra [23]. In Fig. 12.7c we show the pioneering N2 K-shell spectrum recorded in 1973 [21], which used electron impact ionization to create the initial XES core hole state. It clearly shows the signature of the outermost 2σu , 1πu , and 3σg occupied molecular

610

12 Quantum Theory of X-Ray Emission and Thomson Scattering

N2 K-shell x-ray emission (b) Decay configurations

(a) Energy level diagram Relative energies (eV)

3 *u

+9

ionization potential

2p 1 *g

419

~ 0.12 eV 1s-

0

409.8

-8.7

401.1

3 1

u

-15.6 394.2 -16.7 393.1

2

u

-18.8

g

391.0

g

-39

1s 1 1

u g

-409.8 0 -409.9 (c) XES spectrum u

371

1

2

u

1 1

u g

g

u

3 1

u

2

u

1 1

u g

g

3

g

Intensity

2

u

h

2s 2

3 1

391

392 393 394 Emitted photon energy (eV)

395

Fig. 12.7 K-shell x-ray emission in N2 molecules. a N2 energy level diagram and level filling in the ground state. The two compensated spin states are shown in different colors. b Diagram of the electron/hole configurations involved in Kα emission of the 1s core hole, assuming that the active hole is spin-down. Electrons are indicated by filled circles and holes by open circles. The upper configurations with opposite parity 1s holes may decay via dipole and parity allowed transitions to the shown three final state configurations with holes in the 2σu , 1πu , and 3σg orbitals, respectively. The final hole configurations have a long lifetime so that the minimum XES linewidth is determined by that of the 1s core hole of 1s 120 meV [22, 23]. c K-shell emission spectrum, recorded after electron impact ionization with an instrumental resolution of 100 meV by Werme et al. [21], revealing vibrational fine structure

orbitals originating from parity and dipole allowed transitions 2σu → 1σg , 1πu → 1σg , and 3σg → 1σu . Owing to its high instrumental resolution of about 100 meV and the small intrinsic 1s core hole width of 1s = 120 meV, the XES spectrum also reveals the first observation of vibrational fine structure. This has been studied in more detail by use of resonant photon excitation (RIXS), in [27–30]. As discussed in Sect. 13.10.1, the RIXS case needs to be described by second order perturbation theory in contrast to the first order XES case.

12.3 Fundamental X-Ray Emission Experiments

611

12.3.3 L-Shell XES in 3d Metals Denoting the total number of occupied spin orbitals by Ne = 2n e , the polarization independent spontaneous emission width XES , including spin, is obtained as XES =

L Ne 8π 2 ω αf 2 R , 2 λ 3(2L + 1)(2c + 1)

(12.19)

where L = 2 and c = 1. If the decay is stimulated, the incident polarization is preserved since the incident photons n pk clone themselves. The above XES expression corresponds to the polarization averaged absorption width (10.59), which for direct comparison reads 8π 2 ω αf 2 L Nh R . (12.20) XAS = λ2 3(2L + 1) The ratio of the two transition rates is given by Nh XAS = (2c + 1) , XES Ne

(12.21)

where Nh + Ne = 2(2L + 1). The relation (12.21) is consistent with a simple argument. For the 2 p → 1s Kshell case we have L = 1 and c = 0, and for a half filled valence shell, Nh = Ne , absorption and emission are predicted to have the same probability. In the absorption process a core electron is excited to the half empty valence shell. The emission process corresponds to replacing the core electron by a core hole that is excited to the half filled valence shell. For the same radial transition matrix element, the two processes must have the same probability as reflected by (12.21). For the transition metals, the 3d shell gets increasingly filled and the XAS resonance intensities due to 2 p → 3d transitions becomes increasingly weaker from Ti to Ni as shown in Fig. 10.16 reflecting the relative change in the number of valence holes Nh and electrons Ne . The XES spectra of the elementary transition metals, recorded after photon excitation at an energy 30 eV above the L2 resonance energy, are shown in Fig. 12.8 [31].

12.3.4 L3 -Shell XES in Cu Metal As an example we consider in Fig. 12.9 the L3 -shell XAS and XES spectra of Cu metal. In Fig. 12.9a we illustrate in an active electron picture the electronic states involved in the emission process, which complements the absorption case previously shown in Fig. 10.14. We account for the degeneracy and occupation of the upper valence state by assuming that of the total number of 2L + 1 orbitals only the fraction n e /(2L + 1)

612

12 Quantum Theory of X-Ray Emission and Thomson Scattering

Fig. 12.8 L-shell XES spectra of the elementary 3d transition metals [31] (and A. Moewes private commun.)

L-shell XES of 3d transition metals

L3

h

1

= (L2) + 30eV

25

L2 Cu 20

Intensity (arb. units)

Ni Co

15

Fe 10

Mn Cr

5

Ti 0

-20 -15 -10 -5

0

5

10 15 20 25 30

Energy rel. to L3 peak position (eV)

is filled with electrons and can participate in the decay. The lower state of angular momentum c has a hole in the manifold of 2c + 1 core orbitals which can be filled. The two possible spin scenarios are accounted for by multiplying the probability in Fig. 12.9a by a factor of 2. In the following we are interested in the XES spectrum of Cu metal which has a nearly filled 3d shell, described by Nh = 0.5 and Ne = 9.5 [34]. This is reflected by the L-edge XAS spectrum shown in Fig. 12.9b [35] when compared to those of other transition metals in Fig. 10.16. The Cu L-edge XES spectrum is expected to be a signature of the filled 3d density of states (DOS). In practice, one finds that the DOS is better resolved by x-ray photoemission (XPS) than XES. This is shown by comparison of the high-resolution (0.32 eV) XPS spectrum recorded at ω = 1486.6 eV (monochromatized Al Kα radiation) [32] in Fig. 12.9c with the XES spectra in Fig. 12.9d recorded with a spectrometer resolution of 0.8 eV. Although the instrumental resolution in the two measurements was different, there are intrinsic reasons why the photoemission spectra more closely resemble the DOS. In conventional XES, spectra of transition metals contain strong shake-up structure which causes a significant difference of the XES spectra and the DOS. This is

12.3 Fundamental X-Ray Emission Experiments 1

c

valence state |b

nh

orb.degeneracy

ne

# empty orbitals # filled orbitals

Intensity (arb. units)

(a) Emission L=c+1

nh+ne= 2L+1

ne L 3 (2L+1) (2c+1)

core state |a orb.degeneracy 2c +1

613

0.8

L2 0.6 0.4

Intensity

h  =1486.6 eV

0

one empty orbital

EF

L3

0.2

930 940 950 Incident energy (eV)

915 (c) Cu XPS valence band

(b) Cu metal XAS

960

Emission energy (eV) 920 925 930 935 940

0.3 (d) Cu XES valence band

945

h = 932.5 eV 1088.5 eV

0.2

difference

0.1 0 EF

-0.1 -8

-2 0 -4 -6 Binding energy (eV)

-15

5 0 -10 -5 Binding energy (eV)

10

Fig. 12.9 a Angular part of the dipole matrix elements for transitions from valence orbitals of angular momentum L = c + 1 to a single core hole of angular momentum c. We have assumed that a fraction n e /(2L + 1) of valence orbitals is filled with electrons. The emission probability is given in red. b L-shell x-ray absorption spectrum of fcc Cu metal [35]. c High-resolution (0.32 eV) XPS spectrum recorded at ω = 1486.6 eV of the 3d valence band of Cu metal [32]. d Cu valence band recorded by XES with a spectrometer resolution of 0.8 eV after photon excitation with a bandwidth of 0.5 eV to empty states just above the Fermi level (ω = 932.5 eV, thick black line) and excitations to states in the continuum (ω = 1088.5 eV) [33] and M. Magnuson private communication. The spectra were normalized to each other by demanding that their integrated intensity is the same. The difference spectrum reveals the multi-electron shake-up structure present for high energy excitation

illustrated by the black and red XES spectra in Fig. 12.9d, where the core hole was produced with incident x-rays of bandwidth of 0.5 eV that were either tuned just above the L3 threshold (ω = 932.5 eV, black curve) or had an energy ω = 1088.5 eV well above threshold [33]. The shown spectra were normalized to each other so that their integrated intensity is the same. The difference spectrum, shown in blue, reveals the multi-electron shake-up structure present for high energy excitation (also see [36]). The resonantly excited XES spectrum (black) in Fig. 12.9d reveals less details in comparison with the XPS spectrum in Fig. 12.9c. Both spectra reflect the momentum

614

12 Quantum Theory of X-Ray Emission and Thomson Scattering

integrated density of states [33, 37] and are expected to be similar. The shown XPS spectrum is limited by an instrumental resolution of 0.32 eV [32], while the shown XES spectrum is limited by an instrumental resolution of 0.8 eV [33]. However, there are other intrinsic broadening mechanisms in XES. As discussed in Sect. 12.2.6, the XES linewidth is fundamentally limited by the sum of the core hole and final state linewidths  = a + b given by (12.14). For the L3 -edge in Cu the core hole width is a = 0.56 eV according to Table 10.2. The final state consists of a hole in the occupied 3d valence band, similar to XPS, and is expected to have a smaller width b < a . While the intrinsic XPS linewidth is limited by b , the XES linewidth is intrinsically limited by a .

12.4 X-Ray Fluorescence Yield, Linewidths, and Strengths The total angle and polarization integrated spontaneous x-ray emission rate after a core excitation constitutes the fluorescence yield (FY), Yf . For a specific core excitation in a given atom, the fluorescence yield is assumed to be an atomic property, independent of the chemical environment. A recent study of the L-edge FY of Ni suggests that this approximation may only be good to about 30% [38]. Differences of order 20% can also be found in values tabulated in the literature [39–41]. In Fig. 12.10 we have plotted in red (right scale) semi-empirical values of Yf tabulated by Krause [40] for the K-edges of the low-Z atoms C (Z = 6) through Ar (Z = 18) and the L3 -edges of the 3d transition metal atoms Ca (Z = 20) through Ar (Z = 30). For the shown cases Auger decays dominate so that  =  A +  X  A .

12.4.1 Radial Dipole Matrix Element We can use the complementary resonant absorption case in Fig. 10.14 and corresponding emission case in Fig. 12.9a to obtain an empirical value for the radial dipole matrix element R, defined in (10.46). By assuming resonant excitation and adding the active electron to the empty valence shell we obtain from (12.19) X =

L(Ne + 1) 8π 2 ω αf 2 R . 2 λ 3(2L + 1)(2c + 1)

(12.22)

Neglecting the energy difference between filled and empty valence states, which is small relative to the absorption and emission energies, we take ω and λ to correspond to the core hole binding energy. The fluorescence yield is then given by Yf =

L(Ne + 1) X 8π 2 ω αf 2 = R .  λ2  3(2L + 1)(2c + 1)

The radial matrix element is obtained as

(12.23)

615

1

1

-1 A

~ -

10

-2

N

O

F

Na Ne

Mg

Al Si

P

S

Ar

Cl

10

10

-1

-2

C 10

-3

10

-4

Yf =

Yf

X

10

Fluorescence yield Yf

10

-3

X

0

500

1

10

-1

10

-2

10 1000 1500 2000 2500 3000 3500 K-edge binding energy (eV) 1

A

~ -

Ti 10

-3

10

-4

V

Cr

Co Ni Cu Mn Fe

Sc Ca

Yf

-4

10

-1

10

-2

10

-3

10

-4

Zn

Fluorescence yield Yf

L3 shell lifetime widths (FWHM) (eV)

Fig. 12.10 Tabulated fluorescence yields Yf shown in red (right scale) [40] for the K-shells of low Z atoms and L3 -shells for the 3d transition metals. Also plotted (left scale) are the total core hole decay widths  from Table 10.2 (solid black circles) and the dipolar transition widths  X (open circles) derived from the relation  X = Yf 

K-shell lifetime widths (FWHM ) (eV)

12.4 X-Ray Fluorescence Yield, Linewidths, and Strengths

X

300 400 500 600 700 800 900 1000 1100 L3-edge binding energy (eV)

R2 =

λ2  3(2L + 1)(2c + 1) Yf . 8π 2 ω αf L(Ne + 1)

(12.24)

Using the polarization averaged fluorescence yields plotted in Fig. 12.10, we have plotted the radial dipolar matrix elements for the K-shell excitation in the low-Z atoms in Fig. 12.11a. The Z -dependence of R scales approximately as a0 /Z where a0 = 0.0529 nm is the Bohr radius. This is expected since R, given by (10.46), is determined by the spatial overlap with the core orbital. The radial matrix element 2 p → 3d for the L-shell excitation in the transition metal atoms is plotted in Fig. 12.11b. In the figure, the blue squares are calculated according to (12.24) as for the K-edge in (a). Also shown as black circles are the theoretical values from Fig. 10.15. The red diamonds correspond to use of (12.24) with Ne + 1 replaced by Ne + 0.5, which is the so-called transition state model introduced by Slater [42], as discussed in [43]. The three methods agree very well for the heavy 3d metals Mn–Cu, but there are deviations for the elements Ca-Cr, as shown.

616

12 Quantum Theory of X-Ray Emission and Thomson Scattering 8x10 -3 7x10 -3 6x10 -3

(a) K-shell 1s 2p From FY with Ne+1

5x10 -3

(nm)

3x10 -3

Radial matrix element

4x10 -3

1x10 -3

2x10 -3 6 7 8 9 10 11 12 13 14 15 16 17 18 C N O F Ne Na Mg Al Si P S Cl Ar

1.3x10-2 1.2x10-2 1.1x10-2 1.0x10 -2

(b) L-shell 2p 3d From FY with Ne+1 From FY with Ne+0.5 Theory

9x10 -3 8x10 -3 7x10 -3 6x10 -3

20 21 22 23 24 25 26 27 28 29 Ca Sc Ti V Cr Mn Fe Co Ni Cu

Atomic number Z and Element

Fig. 12.11 a Radial transition matrix elements R in nm (1 nm2 = 104 Mb) for the K-shell 1s → 2 p excitation in the low-Z atoms C through Ar, calculated according to (12.24) with c = 0, L = 1 with 4 ≤ Ne ≤ 16 and the values for  and Yf in Fig. 12.10. b Same for the L-shell 2 p → 3d excitation in Ca through Zn. The black circles are taken from Fig. 10.15. For the blue squares we have used (12.24) with c = 1, L = 2 and 0 ≤ Ne ≤ 10. The red diamonds were calculated by substituting Ne + 1 by Ne + 0.5 in (12.24), corresponding to Slater’s transition state model [42]

12.5 Quantum Theory of Thomson Scattering The simplest form of x-ray scattering by atomic matter is Thomson scattering. It was introduced within the classical electrodynamics framework in Sect. 6.2, and its quantum treatment was briefly introduced in Sect. 9.5. The process may be elastic or inelastic and is illustrated in Fig. 12.12 for photon scattering by an atom. The various Thomson scattering processes associated with different energy losses of the incident photons are illustrated in Fig. 12.13. Elastic Thomson scattering is one of the most important x-ray interaction processes with matter since it underlies Bragg diffraction. Inelastic Thomson scattering involves losses in photon energy due to excitations which are illustrated in the figure for solid samples. At the lowest energy ranging up to a few hundred meV, losses are due to excitations of phonons and

12.5 Quantum Theory of Thomson Scattering

617 2

1

1

k1

2

k2 2

q

k1

atom

Fig. 12.12 Scattering geometry. An incident photon of energy ω1 , polarization 1 and wavevector k1 is scattered by an angle 2θ , resulting in an outgoing photon of energy ω2 , polarization 2 , and wavevector k2 . The scattering process transfers an energy ω = ω1 − ω2 and a momentum q = k1 − k2 to the sample

Elastic and inelastic x-ray scattering response of solids elastic

Compton plasmons phonons/ magnons

0

valence electrons

0.5 1

core electrons

10 100 Energy loss (eV)

1000

Fig. 12.13 Schematic overview of the elastic and inelastic scattering response of solids, showing typical excitations due to energy losses of the incident photons

magnons. Above about 1 eV valence electrons may be excited from filled to empty states and near about 10 eV collective excitations of plasmons may be observed. At higher energies core electrons may be excited to unfilled states, and these elementspecific excitations typically overlap with a Compton scattering background. In this book we will not discuss Compton scattering in detail,4 but only show how it becomes increasingly important at large incident photon energies > 30 keV and scattering angles. For more details the reader is referred to [45–47]. In the following we consider in detail how the Thomson scattering of photons is treated in KHD theory. The classical and quantum approaches will be seen to give identical results. This is expected since within the KHD theory, Thomson scattering is a first order interaction process and we have seen in Chap. 5 that the general QED

4

Compton scattering, discovered in 1923 [44] today, has arguably the most important application in radiobiology and radiation therapy.

618

12 Quantum Theory of X-Ray Emission and Thomson Scattering

framework reduces to the classical wave description in first order. Our quantum treatment of Thomson scattering here completes our discussion of first order interactions within the KHD framework, following that of XAS and XES.

12.5.1 Quantum Theoretical Formulation of Thomson Scattering The quantum process of Thomson scattering is described by the energy diagram in Fig. 9.4c. It consists of a non-resonant elastic or inelastic scattering process on the total atomic charge and in space may be pictured as illustrated in Fig. 12.12. In quantum theory, Thomson scattering is described by the first order term in the KHD formula (9.21) given by Wi f =

2π |  f |Hint | i|2 ρ(E f ) δ(E f − Ei ) 

(12.25)

with the interaction Hamiltonian (see (9.22)) given by Hint =

e2 2 e2 A = E 2. 2m e 2m e ω2

(12.26)

As for the x-ray absorption and x-ray emission calculations, the initial and final states |i, and | f  are product states consisting of photon and atomic parts.

12.5.1.1

Evaluation of the Transition Matrix Element

The transition matrix element in (12.25) is evaluated with photon number states, and we denote the incident state as |n p1 k1  and scattered state as |m p2 k2 . They differ in occupation n  = m and may differ in wave vector k1  = k2 and polarization p1  = p2 . As we have done for XAS and XES, we denote the initial electronic state as |a and final state as |b. The complete initial product state consists of a photon and electronic part of the form |i = |a, n p1 k1 , 0 and has a total energy Ei = Ea + n p1 k1 ωk1 . The final state contains one less photon in state |n p1 k1  and a single photon m p2 k2 + 1 = 1 in state |m p2 k2 . It is written | f  = |b, n p1 k1 − 1, 1 and has an energy E f = Eb + (n p1 k1 − 1)ωk1 + ωk2 . The matrix element in (12.25) then takes the form  2    e   f |Hint | i = b, n p1 k1 −1, 1  A2  a, n p1 k1 , 0 . 2m e

(12.27)

12.5 Quantum Theory of Thomson Scattering

619

The vector potential A (or field E) operators are expressed in quantized form according to (3.55) (or (3.60)), and we have (note dot product) 1 e2 e2 2  A = 2m e 4m e 0 uv k k ωk1 ωk2 Vk1 Vk2 1 2   † i(k1 ·r−ωk1 t) −i(k1 ·r−ωk1 t) +  ∗u auk e  u auk1 e 1   † ·  v avk2 ei(k2 ·r−ωk2 t) +  ∗v avk2 e−i(k2 ·r−ωk2 t) .

(12.28)

By inspection of possible matrix elements, one finds that only cross terms, consisting † and their complex conjugates, of pairs of creation and annihilation operators auk1 avk 2 can couple the initial and final photon number states in (12.27). The effective Thomson scattering Hamiltonian therefore consists of such cross terms and is given by HT =

1 e2  4m e 0 uv k k ωk1 ωk2 Vk1 Vk2 1 2   † † ∗ ∗ +  ·  a a  u ·  v auk1 avk ei(k1 −k2 )·r e− i(ωk1 −ωk2 )t . v vk 2 u uk1 2

The only non-vanishing matrix element, which occurs twice, is given by  n p1 k1 −1, 1| a †p2 k2 a p1 k1 |n p1 k1 , 0 =

√ n p1 k1 .

(12.29)

The transition matrix element in (12.27) is obtained as 

12.5.1.2

 b, n p1 k1 −1, 1 |HT | a, n p1 k1 , 0 √ n p1 k1 e2 = [ p1 ·  ∗p2 ] b| ei(k1 −k2 )·r |a. ωk1 ωk2 Vk1 Vk2 2m e 0

(12.30)

The Density of Final States

Next we evaluate the product of the density of states and δ-function It is obtained by use of the photon density expression (3.102) or Vk2 ωk2 2 dk2 /(8π 3 c3 ) and by summing over all electronic final states to the energy conserving δ-function δ(E f − Ei ) = δ(Eb − Ea − ωk1 δ(Eba − ωk1 + ωk2 ). We obtain ρ(E f ) δ(E f − Ei ) =

Vk2 ωk2 2 dk2 8π 3 c3 

δ(Eba − ωk1 + ωk2 ).

in (12.25). ρ(ωk2 ) = |b subject + ωk2 ) =

(12.31)

620

12 Quantum Theory of X-Ray Emission and Thomson Scattering

12.5.1.3

The Thomson Cross Section

We have now all ingredients to write down the Thomson scattering rate (12.25). It may be converted into the double differential Thomson scattering cross section according to 1 d2 Wi f Vk1 d2 Wi f d2 σT = = . dk2 dωk2 1 d dω2 n p1 k1 c d dω2

(12.32)

The differential Thomson scattering cross section is simply the scattering rate Wi f normalized by the incident flux and accounting for the fraction of photons emitted into the solid angle d2 of a detector and its bandwidth d(ω2 ). Inserting the squared matrix element (12.30) and the final state density (12.31) into (12.32), we see that the two photon normalization volumes Vk1 and Vk2 fall out, and we obtain 2

 d2 σT ωk2 e4  = | 1 ·  ∗2 |2  b| ei(k1 −k2 )· r | a δ(Eba −ωk1 +ωk2) 2 2 2 4 dk2 dωk2 16π 0 m e c ωk1 b 2

  2 ωk2 ∗ 2 i(k1 −k2 )· r = r0 | 1 ·  2 | | a δ(Eba −ωk1 +ωk2) . (12.33)  b| e ωk1    b   single electron dynamic structure factor S(k1 −k2 ,ωk −ωk ) 1 2 scattering

Here r0 = 2.82 × 10−6 nm is the classical electron radius or Thomson scattering length, previously defined in (6.17), and we have identified two contributions by underbrackets. The first corresponds to elastic Thomson scattering off a single electron with dimension [area], given classically by (6.19). The second term is called the dynamic structure factor with the sum expressing the collective contributions of all electron states |b. It has the dimension [1/energy] through the δ-function, which conserves the dimensions of the left and right sides of the equation. It is seen to depend on the change in photon energy, ωk1 − ωk2 and momentum, q = k1 − k2 , illustrated in Fig. 12.12. The inelastic formalism remains valid for no change in energy, i.e. the elastic case. We can summarize as follows. The double differential Thomson scattering cross section which describes both elastic (ω2 = ω1 ) and inelastic (ω2 = ω1 ) processes with photon momentum transfer q = k1 − k2 illustrated in Fig. 12.12 into a finite solid angle of detection d2 with a detector bandpass dω2 is given by d2 σT ω2 = r02 | 1 ·  ∗2 |2 S(q, ω1 − ω2 ) d2 dω2 ω1 2 c2 ω2 | 1 ·  ∗2 |2 S(q, ω1 − ω2 ), = αf2 (m e c2 )2 ω1

(12.34)

12.5 Quantum Theory of Thomson Scattering

621

where r02 = 0.079 barn and αf = e2 /4π 0 c 1/137 is the dimensionless fine structure constant which is a fundamental measure of the photon-charge coupling (see (10.40)). The quantity S(q, ω1 − ω2 ) =

   b| eiq·r | a2 δ(Eba − ω1 + ω2 )

(12.35)

b

is the dynamic structure factor of dimension [1/energy] since the δ-function is only defined by its dimensionless integral over energy. The difference ω1 − ω2 corresponds to an inelastic loss with momentum change q = k1 − k2 shown in Fig. 12.12.

12.5.2 Elastic Thomson Scattering: Atomic Form Factor For elastic scattering we have ω1 = ω2 and a = b, so that the differential nonresonant scattering cross section becomes dσT = r02 | 1 ·  ∗2 |2 d2

2  Z     iq·rn  n|e |n = r02 | 1 ·  ∗2 |2 |F 0 (q, Z )|2 . (12.36)   n=1

Here n are the atomic electrons of total number Z , and rn is the distance vector from the nucleus to the n-th electron. The quantity F 0 (q, Z ) is the dimensionless atomic form factor given by (6.25) or 1 F (q, Z ) = − e 0

 ρ(r) eiq·r dr.

(12.37)

At soft x-ray energies where the wavelength λ is larger than the atomic diameter the form factor becomes equal to the number of electrons F 0 (q) = F 0 = Z , and by integration over emission angles (see (6.21)) we obtain for the polarization and emission angle integrated atomic Thomson cross section σT =

8π 2 2 r Z = σe Z 2 , 3 0

(12.38)

where σe = 0.665 barn is the single-electron Thomson cross section. This agrees with the classical result given by (6.28).

622

12 Quantum Theory of X-Ray Emission and Thomson Scattering

12.5.3 Inelastic Thomson Scattering: Dynamical Structure Factor The dynamic structure factor S(q, ω) =

   b| eiq·r | a2 δ(Eba − ω1 + ω2 )

(12.39)

b

contains all atomic matter dependent information. It describes the excitation spectrum of the sample as a function of transferred momentum q = k1 − k2 and energy change ω1 − ω2 . The term “dynamic” reflects the fact that it may also be written in terms of a time dependent two-particle density correlation function. It then expresses the fluctuation-dissipation theorem that connects the dissipative response of a system to a perturbation with the equilibrium fluctuations of the system [46]. One would expect that the scattered intensity peaks at the elastic scattering condition ω1 = ω2 and q = 0 gradually become weaker with increasing energy and momentum. This is seen by expanding the operator which couples different electronic states according to 1 (12.40) eiq·r = 1 + iq · r + (iq · r)2 + .... 2 The first monopole excitation term describes elastic forward scattering q = 0 where electrons oscillate with the x-ray frequency and re-emit radiation of the same frequency in the same direction. The second term is the electric dipole term often called the Raman scattering term and corresponds to dipole excitations between states a and b, involving an energy transfer ω1 − ω2 .5 The third term corresponds to magnetic dipole and electric quadrupole excitations as discussed in Sect. 11.4, and in accordance with (11.11) can be written as i 1 (iq · r)2 = (r × p) · (q ×  p ) 2 2   magnetic dipole

i + (q · r)(p ·  p ) + (q · p)(r ·  p ) . 2  

(12.41)

electric quadrupole

Another way to derive the different excitations associated with S(q, ω1 − ω2 ) [48] is to expand the scattering operator in terms of spherical harmonics or spherical tensors according to

5 Note that for absorption and emission, the dipole approximation corresponds to approximating the exponential by 1 as in (10.5), while in Thomson scattering it corresponds to the second term on the right in (12.40).

12.5 Quantum Theory of Thomson Scattering

eiq·r = 4π

l ∞

623

∗ il jl (qr ) Yl,m (eq ) Yl,m (er )

l=0 m=−l

=

∞

l

il (2l + 1) jl (qr ) Cm(l)∗ (θq , φq ) Cm(l) (θr , φr ),

(12.42)

l=0 m=−l

where jl (qr ) are Bessel functions given for l = 0, 1, 2 by (10.21), and the unit vectors eq and er define the coordinate frames of the position vector r and momentum transfer vector q. Note that the scattering operator depends on both the direction (through the angle dependent functions Yl,m ) and the magnitude (through jl (qr )). For a dipole transition, for example, the q-vector assumes the role of the E-vector in the determination of the angular dependence of the transitions and maximum inelastic excitations occur along q.

12.5.4 Core Shell Excitations: X-Ray Raman Scattering (XRS) Inelastic Thomson scattering also occurs when the photon energy loss is due to core electron excitations. The process is similar to the loss in energy of high energy electrons in an electron microscope referred to as electron energy loss spectroscopy [49] or when used for core excitations as inner shell electron energy loss spectrosocopy (ISEELS) [43, 50]. In the limit of small momentum transfer, where the dipole approximation holds, inelastic loss spectra recorded with electrons or photons are equivalent to XAS spectra. The long history of inelastic x-ray spectroscopy has been reviewed by Bergmann, Glatzel, and Cramer [51]. Following the modern theoretical description by Mizuno and Ohmura [52] and its clear experimental demonstration by Suzuki [53] in 1967, it is often referred to by the name x-ray Raman scattering (XRS). Another name is non-resonant inelastic x-ray scattering (NIXS) [46]. In using the acronym NIXS we have to keep in mind that the “N” for “non-resonant” derives from the historical view that Thomson scattering is a first order “non-resonant” process that in its elastic form involves the entire atomic charge. When inelastic excitations of specific core shells are picked out by their characteristic loss energies, inelastic Thomson scattering can nevertheless provide information about resonant excitations involving specific atomic subshells. XRS or NIXS is typically performed with incident hard x-rays around 10 keV and use of a crystal monochromator to select an incident energy bandpass, and the scattered radiation is analyzed by a crystal spectrometer. Experiments may be performed by scanning either the incident, ω1 , or scattered, ω2 , photon energy. The measured XRS energy loss extends from the characteristic XAS threshold of interest to larger values as illustrated for the C K-edge of graphite in Fig. 12.14 [51]. The spectrum was recorded as specified in the caption.

624

12 Quantum Theory of X-Ray Emission and Thomson Scattering 4

10

3

10

Graphite K-shell

Relative intensity

Relative intensity

10

Elastic peak

XRS

XAS

2

Compton peak 10

320 300 Energy (eV)

340

XRS h

1

280

2 =6460eV

6900 6500 6700 Incident photon energy h 1 (eV) 0

100

200 300 400 500 Energy loss (eV)

7100 600

Fig. 12.14 Experimental inelastic photon energy loss spectrum of the carbon K-shell in graphite −1 [51], recorded with an instrumental resolution of 2.5 eV and a range of momentum transfers 4Å ≤ −1 q ≤ 6Å . The loss spectrum reveals the elastic peak, the Compton scattering background peak, and the XRS loss. In the measurement the scattered photon energy was detected at a fixed energy ω2 = 6460 eV and the incident energy ω1 was scanned as indicated. The XRS spectrum in the inset was recorded with an instrumental resolution of 1 eV, while the XAS spectrum was recorded by electron yield detection with a resolution of 0.15 eV

In practice, losses associated with core electron binding energies (x-ray absorption edges) larger than about 1 keV become too weak. For this reason, most applications have involved losses associated with soft x-ray core excitations such as the C K-edge around 285 eV and O K-edge around 535 eV as reviewed in [46, 54–56]. For the study of low-Z elements like C and O, the dipole approximation is easily satisfied in practice. The 1s core shell radius r1s  is to a good approximation given by the radial dipole matrix element R1s plotted in Fig. 12.11a which scales with Z approximately as a0 /Z , where a0 = 0.0529 nm is the Bohr radius. For the O K-shell, for example, we have r1s  R1s 5 × 10−2 Å. The momentum transfer is given by q 2 = [ω12 + ω22 − ω1 ω2 cos(2θ )]/c2 , where θ is the angle between incident and scattered photons (see Fig. 12.17). At a relatively large scattering angle of 2θ = 90◦ , for example, and an incident energy of 10 keV we have a momentum transfer of about q 7Å−1 . This yields r1s q = 0.35, and the dipole approximation still holds. In this case, XRS gives results similar to XAS spectroscopy as shown in the inset of Fig. 12.14 and also demonstrated in [54, 55].

12.5 Quantum Theory of Thomson Scattering

12.5.4.1

625

Optimum Range of Momentum Transfer in XRS

At finite but small wavevector transfer q < 5Å−1 , the XRS cross section is dominated by the dipole term ∝ q · r in the expansion (12.40) or (12.42) of the matrix element of the dynamic structure factor. In order to illustrate the relative sizes of the monopole ( = 0), dipole ( = 1), and quadrupole ( = 2) terms in (12.42), we have calculated the expectation values ∞ R1s | j (qr )|R2 p  =

R1s j (qr ) R2 p r 2 dr

(12.43)

0

corresponding to 1s → 2 p excitations in oxygen atoms (Z = 8). We used the hydrogenic wavefunctions given by (10.12) and the Bessel functions j (qr ) for  = 0, 1, 2 2 2 r and the correspondgiven by (10.21). The results for the radial probabilities Rn ing multi-pole terms (12.43) are shown in Fig. 12.15, neglecting interference terms between the channels. The multi-pole terms shown in Fig. 12.15 only account for the magnitudes, q, but not the directions, q, of the scattered photons. The monopole term only contributes to the elastically forward scattered intensity around q = 0, since inelastic scattering involves excited states that are orthogonal to the ground state, and can only be reached through the higher order multi-pole terms. We see that for relatively small momentum transfer q ∼ 5 Å−1 , the dynamic structure factor may be approximated by the dipolar  = 1 form

| b| eq · r |a|2 δ(Eba − ω1 + ω2 ), (12.44) S(q, ω) = q 2 b

where eq is a unit vector in the direction of q. There is another important reason for keeping the momentum transfer relatively small, i.e. q < 5 Å−1 . It facilitates removal of the Compton scattering background that appears as a broad background in the inelastically scattered intensity, as seen

2 r 2 for oxygen atoms calculated with hydrogenic wavefunctions Fig. 12.15 Radial probabilities Rn (10.12) and q-dependence of multi-pole terms (12.43) calculated with the Bessel functions j (qr ) for  = 0, 1, 2 given by (10.21). Note that the monopole term does not contribute to inelastic scattering

626

12 Quantum Theory of X-Ray Emission and Thomson Scattering

in Fig. 12.14. The change of the Compton background with momentum transfer is shown in more detail in Fig. 12.16 in conjunction with K-shell XRS spectrum of a diamond powder [57]. With increasing q the Compton peak shifts to larger loss energies [56]. For very large momentum transfer q, XRS evolves into Compton scattering [45, 56] and the XRS onset for a core shell excitation may be viewed as the threshold of its Compton profile as illustrated in Fig. 12.16 [57].

12.5.4.2

XRS Versus XAS Cross Sections

In the following we are interested in a quantitative comparison of the relative size of the XRS and XAS cross sections. We will see that the XRS cross section is only of order 10−10 of the corresponding soft x-ray absorption cross section, so that XRS studies greatly benefit from state-of-the-art synchrotron facilities or XFELs. Nevertheless, in some cases the cross-section disadvantage is acceptable because of the following unique capabilities of XRS. For the study of absorption fine structure in the soft x-ray region up to about 1 keV, x-ray Raman scattering or XRS may be used instead of conventional XAS. While the XAS cross section is significantly larger, XRS offers increased penetration lengths owing to the use of hard x-rays. This avoids the vacuum environment required for soft x-ray XAS and facilitates transmission studies of samples such as gases, liquids, and bulk materials and their investigations in extreme conditions such as at high pressure and temperature.

XRS and Compton peaks for C K-shell in diamond

Scattered Intensity (arb. units)

Fig. 12.16 q-dependence of the Compton background underlying XRS, illustrated by study of the C K-edge of diamond powder [57]

XRS

Compton -1

q =14.6A

0

-1

q =11.7A

0 -1

q = 5.7A

0 0

200

800 1000 1200 600 400 Energy loss (eV)

12.5 Quantum Theory of Thomson Scattering Fig. 12.17 Scattering geometry for non-resonant Raman scattering, indicating the wavevector and polarization dependence in the inelastic scattering process

627

y 1 1

x

k1

^

2

k1

k2

2

sample

^

2 = ( k 2 x 1) x k 2

q

z

For our quantitative description of XRS we envision the scattering process as schematically shown in Fig. 12.17. For incident linear polarization the polarization vectors are real and for charge scattering the scattered polarization direction is  2 = (kˆ 2 × 1 )× kˆ 2 , and with the notation of Fig. 12.17 we have | 1 ·  2 |2 = cos2 (2θ ) when q lies in the x–z plane and | 1 ·  2 |2 = 1 when q lies in the y–z plane. When the dynamic structure factor is written in the dipolar form (12.44), the state |a in the matrix element is the electronic ground state, and there is still a sum over all atomic shells |b. If we energy select a specific excitation |a → |b where |b is a specific core hole state, we obtain an element-specific XRS cross section which describes the core hole production in a specific atomic shell. By selecting a specific final core hole state |b, we need to also consider its lifetime or decay width b . We may then replace the δ-function by a Lorentzian which integrates over energy to the same unit value (assuming a single final state) as the δ-function. The Lorentzian is given by (10.36) and has the dimension [1/energy]. The double differential cross section then becomes d2 σXRS ω2 2 (b /2)2 = r02 q 2 | 1 ·  ∗2 |2 | b| eq · r |a|2 . d2 dω2 ω1 π b (Eba − ω1 + ω2 )2 + (b /2)2

(12.45) This may be directly compared to the XAS cross section written in a similar form by use of (10.38) and (10.40) as σXAS =

(b /2)2 8π ω αf |b| 1 · r|a|2 . b (Eba − ω)2 + (b /2)2

(12.46)

We see that apart from the unit polarization vector the squared marix element is the same in both cases. The difference in cross section is therefore simply given by the prefactors.

628

12 Quantum Theory of X-Ray Emission and Thomson Scattering

12.5.5 Example: Typical O K-shell Cross Sections For a comparison of the XRS and XAS cross section we consider the case of the O K-shell and neglect polarization effects. For XRS we assume monochromatic high energy incident photons of energy ω1 10 keV and bandwidths less than the O Kshell core hole width of b 0.15 eV, and we measure the scattered photon intensity at an energy around ω2 9.5 keV corresponding to the O K-edge loss. We assume a momentum transfer of q ∼ 5 Å−1 where the Compton peak lies below the K-shell loss spectrum. With r02 = 0.079 barn= 0.079 × 10−10 nm2 = 7.9 × 10−10 Å2 we obtain a prefactor value in (12.45) of r02 q 2

ω2 2 8.4 × 10−8 . ω1 π b

(12.47)

For the XAS case we assume incident photons around ω = 535 eV, and with a core hole width of b 0.15 eV we obtain the prefactor in (12.46) as 8π ω αf 6.5 × 102 . b

(12.48)

The relative size of the differential XRS and XAS cross sections is, therefore, d2 σXRS d2 dω2

σXAS

1.3 × 10−10 .

(12.49)

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

A. Einstein, Phys. Z. 18, 121 (1917) A. Einstein, Verh. Deut. Phys. Ges. 18, 318 (1916) M. Planck, Verh. Deut. Phys. Ges. 2, 202 (1900) M. Planck, Verh. Deut. Phys. Ges. 2, 237 (1900) J. Stöhr, Phys. Rev. Lett. 118, 024–801 (2017) J. Stöhr, Adv. Optics Photonics 11, 215 (2019) V. Weisskopf, E. Wigner, Z. Physik 63, 54 (1930) V. Weisskopf, E. Wigner, Z. Physik 65, 18 (1930) D.V. Strekalov, A.V. Sergienko, D.N. Klyshko, Y.H. Shih, Phys. Rev. Lett. 74, 3600 (1995) Y. Shih, An Introduction to Quantum Optics: Photon and Biphoton Physics (CRC Press, Boca Raton, Florida, 2011) W. Voigt: Sitzungsbericht der Bayerischen Akad. Wissenschaften p. 25 (1912) O. Björneholm, A. Nilsson, A. Sandell, B. Hernnäs, N. Mårtensson, Phys. Rev. Lett. 68, 1892 (1992) W. Wurth, P. Feulner, D. Menzel, Phys. Scripta T41, 213 (1992) R. Röhlsberger, DESY, private communication. For a review of state-of-the-art Mössbauer spectroscopy see R. Röhlsberger, Fortschr. Phys. 61, 360 (2013)

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Chapter 13

Quantum Theory of X-Ray Resonant Scattering

13.1 Introduction and Overview The goal of this chapter is to extend the first order KHD formalism describing x-ray absorption, emission, and Thomson scattering to the description of second order phenomena, most prominently encountered in resonant x-ray scattering. The general KHD formula (9.21) contains both Thomson (first order) and resonant scattering (second order) contributions, and they may even interfere because the respective scattering lengths need to be added first before the total scattered intensity is calculated as the absolute value squared.1 In this chapter we specifically discuss the second order term in the KHD formula (9.21) which entails the quantum theory of both energy dependent elastic (REXS) and inelastic (RIXS) processes. The theory will be illustrated through experimental examples. For simplicity and clarity we will omit the Thomson scattering term discussed in Sect. 12.5, which may be envisioned to contribute a polarization dependent but energy independent “background”. There are many advantages of using resonant scattering rather than conventional Thomson scattering. The most important ones are • Elastic resonant scattering (REXS) is typically used for coherent diffractive imaging, while inelastic scattering (RIXS) is used for spectroscopic studies of the electronic and spin structure of the valence electrons, including its energy/momentum dispersion. • RIXS differs from XES spectroscopy through the link of the excitation and emission processes and reduces to XES when the excitation process is non-specific, as for broadband incident x-rays. 1 Interference between first and second order scattering amplitudes exists when the final states of the Thomson and resonant terms are the same. An example is a purely elastic scattering event where the final state is not only the electronic but also the vibrational ground state in a molecule. For this case, the polarization dependent interplay between the Thomson and resonant term can be seen from Fig. 1(c) of Phys. Rev. Lett. 106, 153004 (2011).

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_13

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• The intrinsic spectral resolution or measured linewidth of resonant scattering is not limited by the core hole widths of the intermediate states through which the electronic system passes from the initial to the final state. This leads to a significant advantage of RIXS over conventional XES. • Through its dependence on the incident photon energy, resonant scattering is element and chemical state specific. Like Thomson scattering, the resonantly scattered intensity depends on polarization, momentum transfer, and energy loss. • Resonant scattering offers enhanced cross sections over Thomson scattering and through its polarization dependence can separate charge and spin dependent processes. Spin dependent resonant scattering is much larger than spin dependent Thomson scattering. • The resonant process links electronic excitation and decay processes that both obey dipole and parity selection rules, resulting in the selection of specific final states that cannot be reached in a single dipole transition from the electronic ground state. In other cases, the final states may be a subset of those that are reached in conventional XES spectroscopy. With the advent of brighter x-ray sources, RIXS has attracted considerable attention in recent years because of its capability of providing unique information. RIXS is complementary to valence band photoemission and inelastic neutron scattering, yielding information about the low energy electronic and spin excitations in solids as previously illustrated in Fig. 12.13. We mention upfront that resonant x-ray scattering which specifically measures the scattered photons has complementary analogues in resonant photoemission and Auger spectroscopies as reviewed in [1, 2]. We will not discuss the latter techniques which are based on the measurement of electrons, keeping with the “x-ray spirit” of this book.

13.2 Formulation of Resonant Scattering: REXS and RIXS Resonant scattering, including both elastic, REXS, and inelastic, RIXS, processes, is described by the second order term in (9.21) which in the dipole approximation reads  2 2πe2   f |r·E− |mm|r·E+ |i  res W i→ f =   ρ(E f ) δ(E f − Ei ). (13.1)    m E i − Em Again, the states |i, |m, | f  are products of photon number states and electronic states. We shall label the latter |a, |c, |b, respectively. The KHD formula shows that second order processes consist of two consecutive and linked first order XAS and XES processes. This “link” between the probability

13.2 Formulation of Resonant Scattering: REXS and RIXS

633

amplitudes of the two consecutive excitation and decay processes distinguishes it from first order XES. The second order process consists of a sum over all energetic paths through intermediate states m that link the initial and final states before the expectation value is formed as the absolute value squared. If different intermediate core hole states are involved, paths can interfere and such interference effects most clearly distinguish second order from two consecutive first order processes.

13.2.1 Evaluation of the Double Matrix Element The resonant scattering rate in (13.1) is calculated by assuming that the incident flux may in general contain photons in two different modes p1 k1 and p2 k2 with corresponding energies ω1 and ω2 . The case where all photons are in the same mode is a special case of the general formalism. The photons np1 k1 excite the electronic system from the ground state |a of energy Ea to intermediate electronic states |c of energy Ec . The photons np2 k2 may then de-excite the system from states |c to the final state |b of energy Eb . The process is elastic if ω1 = ω2 so that |b = |a and inelastic if ω2 < ω1 so that the system ends up in an excited state with Eb > Ea . The photon parts of the matrix elements in (13.1) are again evaluated by the action of the destruction and creation operators on number states as done in (10.2) and (12.2). According to (3.61), the absorption operator E+ ∝ a p1 k1 destroys photons in the state p1 k1 and the emission operator E− ∝ a †p2 k2 creates photons in the state p2 k2 . The matrix elements in (13.1) are evaluated by considering the states involved. We have an initial state (13.2) |i = |a, n 1 , n 2  with an energy Ei = Ea + n 1 ω1 + n 2 ω2 . The final state is | f  = |b, n 1 − 1, n 2 + 1,

(13.3)

with energy E f = Eb + (n 1 − 1)ω1 + (n 2 + 1)ω2 . We can now find the intermediate states which give non-vanishing photon matrix elements in (13.1). There are only two kinds. First, (13.4) |m = |c, n 1 − 1, n 2  with energy Em = Ec + (n 1 − 1)ω1 + n 2 ω2 . In this case the energy denominator in (13.1) is given by Ei − Em = Ea − Ec + ω1 . The second allowed intermediate state is virtual and given by |m = |c, n 1 , n 2 + 1

(13.5)

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13 Quantum Theory of X-Ray Resonant Scattering

with energy Em = Ec + n 1 ω1 + (n 2 + 1)ω2 . This virtual state contains two more photons than the other intermediate state. Now the energy denominator in (13.1) is given by Ei − Em = Ea − Ec − ω2 . By explicit insertion of the states and energies for the two pathways and by making the dipole approximation, which eliminates the k dependence of the transition operator, the resonant scattering rate per solid emission angle and per emitted energy band width is obtained as [3, 4], ω 2 ≤ω1 res  ω1 ω23 2 αf2  W a→b np2 k2 +1 = 1 2 d2 dω2 c b    b|r ·  p1 |cc|r ·  ∗p2 |a 2 b|r ·  ∗p2 |cc|r ·  p1 |a   × −  ω1 − (Ec − Ea ) + i c /2 ω2 + (Ec − Ea ) c

× δ[(Eb −Ea )−(ω1 −ω2 )] .

(13.6)

Here 1 is the incident photon flux (see (10.3)). This expression is valid for elastic as well as inelastic scattering. The important factor n p2 k2 + 1 represents stimulated x-ray emission by the incident photons in state |n p2 k2  with energy ω2 . When Ea = Eb , the scattering is elastic, and when Eb = Ea , the scattering process is inelastic. The finite lifetime of the final state is neglected since energy conservation is simply assured by the Dirac δ-function between the initial state |a and final states |b, independent of the intermediate core hole states |c. Note that the δ-function is only defined through its integration over energy which yields a unitless number, so that it formally has the dimension [1/energy]. The sum over intermediate electronic states |c expresses paths through which the quantum mechanical system evolves from |a to |b. We have inserted the factor i c /2 in the denominator of the first term in the sum over intermediate states c to account for its finite decay width c . It also eliminates a singularity for ω1 = Ec − Ea . The second term in the sum over intermediate states is non-resonant since the denominator is always large. The intermediate states |c may be real or virtual, as illustrated in Fig. 13.1 for different cases. Equation (13.6) includes two “resonant” terms that are distinguished by the denominators. In practice, there are two situations determined by the relative size of ωi and the average resonance energy E0 Ec − Ea , since the natural decay energy width c /2 is typically small. The most common situation corresponds to ωi ≈ E0 . The transition rate is then determined by the larger first term, since ω1 − E0 in the denominator is small while ω2 + E0 is large. The second resonant term can then be omitted, referred to as the rotating wave approximation. The case ωi  E0 shown in Fig. 13.1a, labeled “Thomson scattering”, corresponds to transitions through virtual states of much higher energy than any true resonant states. In this case both terms in (13.6) contribute. As discussed by Loudon [5], this corresponds to a description of Thomson scattering through a second order process where the usual Hamiltonian e2 A2 /(2m e ) in (9.22) is replaced by the photoelectric Hamiltonian e p ·A/m e . However, this description of Thomson scattering as a

13.2 Formulation of Resonant Scattering: REXS and RIXS

(a) “Thomson Scattering”

635

(c) Resonant Scattering

(b) Rayleigh & Raman Scattering

|c

|c i

f

i

f

i

f

|c |a

|b

|a

|b

|a

|b

Fig. 13.1 Illustration of second order scattering processes in the optical and x-ray regime expressed by (13.6). The shown states |a and |b are the initial and final electronic states. The intermediate states |c may be real or virtual as discussed in the text. As discussed in the text, the description of “Thomson scattering” through a second order process is somewhat problematic

second order process is problematic since it does not properly converge to the conventional Thomson scattering cross section derived in Sect. 12.5 at high x-ray energies. Illustrated in Fig. 13.1b is the case of Rayleigh and Raman scattering in the optical range. Since air molecules have larger electronic energy differences than the energy of optical light, the scattering of sun light by the earth’s atmosphere is dominated by virtual transitions through lower energy states. Rayleigh scattering is elastic Eb = Ea , while Raman scattering is inelastic. The Raman process allows observation of states |b that cannot be reached directly by dipole transitions |a → |b because of selections rules. In resonant scattering, illustrated in Fig. 13.1c, the intermediate states |c are real electronic states through which the system passes to the final state |b. In the shown configuration picture (see discussion below in Sect. 13.2.2 and Fig. 13.2), the initial state |a is the electronic ground state, and in the x-ray regime the intermediate state |c consists of an excited electron in an empty valence state and a hole in a core state. The final state |b is reached by an electronic decay, which may be envisioned by the electron decaying into the core hole or the core hole bubbling up to the filled valence shell. The final state |b is an excited electronic state whose energy is higher than that of the ground state |a. In x-ray science Eb − Ea ranges from meV vibrational excitations to electronic excitations within the valence shell up to a few eV. By introducing the unit vector rˆ and factoring the radial matrix element R, we obtain from (13.6) in the dipole and rotating wave approximations res  4π 2 ω1 ω2 αf2 4  W a→b = 1 R np2 k2 +1 d2 dω2 λ22 2  ω 2 ≤ω1   b|ˆr ·  ∗p2 |cc|ˆr ·  p1 |a   ×    c ω1 − (Ec − Ea ) + i c /2  b

× δ[(Eb −Ea )−(ω1 −ω2 )] .

(13.7)

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13 Quantum Theory of X-Ray Resonant Scattering

Transition rates W may be expressed in terms of the corresponding cross sections σ by dividing through the incident photon flux , yielding the following relations for the integrated, differential, and double differential scattering cross sections σ W =  σ,

dσ W dσ W = , = . d d d dω d dω

(13.8)

The differential scattering cross section is then obtained from (13.7) as follows. The general expression for the double differential resonant cross section, which describes elastic (REXS) and inelastic (RIXS) x-ray scattering, is given in the dipole and rotating wave approximations by res  dσa→b 4π 2 ω1 ω2 αf2 4  = R np2 k2 +1 d2 dω2 λ22 2  ω 2 ≤ω1   b|ˆr ·  ∗p2 |cc|ˆr ·  p1 |a   ×    c ω1 − (Ec − Ea ) + i c /2  b

× δ[(Eb −Ea )−(ω1 −ω2 )] .

(13.9)

The states |a, |b, |c are electronic states of the sample and may include their vibrationally split substates. The Dirac δ-function assures strict energy conservation between the initial state |a and final state |b that does not involve the intermediate states |c. It has the important consequence that the intrinsic (i.e. in the absence of instrumental contributions) REXS and RIXS linewidth of the scattered photons is not limited by the finite intermediate state widths c , as first pointed out by Weisskopf in 1931 [6].

13.2.2 One-Electron Versus Configuration Picture Figure 13.2 illustrates different ways to picture elastic and inelastic resonant scattering for the case of 2 p ↔ 3d scattering. On the left we show how resonant scattering is pictured in a one-electron model by extension of the x-ray absorption and emission picture. This model follows the motion of the “active” electron from the 2 p core to the 3d valence shell and back to the core. If the same electron is involved, we have elastic scattering, if an electron of different energy is involved in the de-excitation process, the scattering is inelastic. As illustrated in the middle, one can view resonant scattering also by considering the motion of a hole with the energy diagram turned upside down. On the right we show the proper configuration picture which corresponds to Fig. 13.1. It clearly shows that

13.2 Formulation of Resonant Scattering: REXS and RIXS One-electron (hole) model

2p

1

2p

Configuration model

hole motion elastic inelastic

electron motion elastic inelastic

3d

1

1

637

|c

1

2

3d

1

1

2

3d n+12p 5

1

|a

3d

n

2p6

2

[ 3d n ]*2p6

|b

Fig. 13.2 Resonant elastic and inelastic x-ray scattering in different models. On the left we show how in a one-electron model we follow the motion of the “active” electron from the 2 p core to the 3d valence shell and back to the core. If the same electron is involved, we have elastic scattering, and if in the de-excitation process an electron of different energy is involved, the scattering is inelastic. The same process can also be pictured by considering the motion of a hole as shown in the middle. The energy diagram is turned upside down. On the right we show the proper configuration picture where all three states are products of core and valence states

the core hole state is the intermediate excited state |c through which the system passes from the initial state |a to the final state |b. There are two important messages in the configuration picture which will emerge from our later detailed treatments, but we shall already highlight them in a box. The core hole intermediate state |c has no influence on energy conservation between the initial and final states. Therefore the core hole energy width does not limit the energy resolution of the resonant scattering process |a → |b. The paths through the intermediate core hole state |c allow final states |b to be reached that are not accessible through direct dipole transitions |a → |b. This is similar to optical Raman scattering [7].

13.2.3 Coherent Second Order Versus Consecutive First Order Processes In the following we shall adopt the electronic configuration picture and label the ground state |a, the intermediate state |c, and final state |b, as shown on the right of Fig. 13.2. We consider two intermediate states |c1  and |c2  to illustrate their interference contribution. We assume that the two states are separated by = Ec2 − Ec1 > 0 and have the same natural decay width . For convenience, we assume linear polarization so that the transition matrix elements are real and describe the dependence on the incident photon energy ω1 in terms of the detuning energy from

638

13 Quantum Theory of X-Ray Resonant Scattering

the upper state is |c1  as c1 =  = ω1 − (Ec1 − Eca ) so that the detuning energy from the upper state is c2 =  − . The absolute value squared of the double matrix element in (13.9) then takes the form     b|ˆr ·  |mm|ˆr ·  |a 2 p2 p1     m=c .c ω1 − (Em − Ea ) + i /2  1

=

2

|b|ˆr ·  p2 |c1 |2 |c1 |ˆr ·  p1 |a|2 |b|ˆr ·  p2 |c2 |2 |c2 |ˆr ·  p1 |a|2 + 2 2  + /4 ( − )2 + 2 /4 

 separate excitation and decay processes

+2 

b|ˆr ·  p2 |c1 c1 |ˆr ·  p1 |ab|ˆr ·  p2 |c2 c2 |ˆr ·  p1 |a /4 2 ( − )+ (− )+ 2 /4 + /4

2

2

.

(13.10)



interference of the two processes

The first underbrace identifies separate paths through the two intermediate states caused by vibrational or electronic spilitting. The second underbrace expresses interference of the paths. The existence of interference is a consequence of a fundamental rule of quantum mechanics, most clearly articulated and emphasized by Feynman [8, 9]. In a process between well-defined initial and final states the probability amplitudes of all alternative indistinguishable paths must first be summed before the total probability is calculated as the absolute value squared. In diffraction, for example, we have to sum (integrate) over all possible spatial paths of photons (or electrons), as was utilized in the quantum derivation of the van Cittert–Zernike theorem in Sect. 4.7. In REXS/RIXS we have to instead sum over all energetic paths of photons through electronic intermediate states. Equation 13.10 expresses a fundamental rule of quantum mechanics, most clearly articulated and emphasized by Feynman [8, 9]. When considering a process between well-defined initial and final states, probability amplitudes of all alternative paths are first summed before the total probability is calculated as the absolute value squared. In diffraction, for example, we have to sum (integrate) over all possible spatial paths of photons (or electrons), as was utilized in the quantum derivation of the van Cittert– Zernike theorem in Sect. 4.7. In REXS/RIXS we have to sum over all energetic paths of photons through electronic intermediate states. When interference between different intermediate state paths is neglected, the general case reduces as follows      b|ˆr ·  ∗ |cc|ˆr ·  p |a 2   b|ˆr ·  ∗p |cc|ˆr ·  p1 |a 2 1 p2   2    −→   ω − (E − E ) + i /2  . (13.11)  c ω1 − (Ec − Ea ) + i /2  1 c a c 

 

 includes interference

no interference

13.2 Formulation of Resonant Scattering: REXS and RIXS

639

The second order KHD formalism, which corresponds to an inseparable coherent excitation/decay process on the left, is approximated on the right by independent consecutive first order KHD processes. The presence of interference terms changes the spectral shape of RIXS spectra and interference of vibrational amplitudes also affects the quasi-elastic REXS lineshape. This will be illustrated by the REXS/RIXS spectra of N2 and O2 below.

13.2.4 REXS/RIXS Terminologies The complicated general resonant term (13.10) has led to much discussion in the literature following the pioneering theoretical work of Gel’mukhanov et al. [10] and Åberg and Tulkki [3, 4, 11] around 1980. By now the theory has been applied to matter in various forms and compared to experiments. The extensive work in the atomic and molecular (AMO) community has been reviewed by Gel’mukhanov et al. [2, 12]. It is interesting that it has taken three to four decades for synchrotron experiments to achieve similar resolution (∼ 100 meV) as the early AMO XES studies by the Uppsala group in the 1970s, represented by the spectra of Ne in Fig. 12.6 and N2 in Fig. 12.7. While early XES studies used electron excitation of gases and grating spectrographs operated in higher order, synchrotron-based photon in/out experiments long suffered by intensity/resolution trade-offs which were overcome only with third generation x-ray sources and large dedicated grating-based spectrometers [13–17]. At present, the best soft x-ray RIXS instruments have a resolution of about 30 meV at photon energies extending to about 1 keV [18]. In the hard x-ray region around 10 keV, instrumental bandwidths of 4 meV have been achieved in RIXS spectrometers based on flat-quartz crystal optics [19]. Early AMO studies have been extended over the years to various forms of matter. By now RIXS has also been utilized for the study of polymers [20, 21], chemisorption systems [22], metal complexes [23–25], solids with weak and strong valence correlations [26–29], and solids under high pressure [30]. The rich landscape of RIXS and its utilization by different communities has also led to the use of different approximations of the general KHD formalism, resulting in terminologies that we will briefly outline below. Today the generic terms “x-ray emission spectroscopy”, XES, or “fluorescence spectroscopy” are mainly used when the radiative decay follows a non-specific creation of core holes, as by high energy or broad bandwidth x-rays or by ionizing electrons. The modern use of resonant excitations has led to a distinction of the REXS/RIXS processes based on the role of the intermediate states through which the system passes from the ground to the final state. In particular, one considers whether the photon probability amplitude or wavefunction passes through the allowed intermediate core hole states with or without interference of the paths, as expressed by (13.11). This picture closely resembles Feynman’s quantum formulation of the double-slit exper-

640

13 Quantum Theory of X-Ray Resonant Scattering

iment (see Sect. 8.5.1), with the spatial (diffraction) paths through the slits replaced by energetic (spectroscopy) paths through two or more intermediate states. In RIXS, the inclusion of the interference of all energetic paths, which is a natural consequence of their indistinguishability in quantum mechanics, has been denoted by the terms “one-step RIXS” [31, 32] or “coherent inelastic scattering” [33, 34]. When interference between the paths is neglected, the “up” and “down” processes may be viewed as distinguishable, decoupled or incoherent. This approximation expressed by (13.11) has led to the terms “incoherent inelastic scattering” [33, 34] or “two-step RIXS” [12, 31, 32]. The term “resonantly excited XES” has also been used [22, 35]. Note, however, that this case still differs from XES and only merges into it in the limit that the incident photon energy has a broad bandwidth or is well above the emission energy so that the first excitation step is essentially non-resonant. In the condensed matter community, where much emphasis is placed on the study of correlated materials, the names “direct RIXS” and “indirect RIXS” have been introduced [27, 29, 36, 37]. These terms are not linked to the one- versus two-step distinctions above but break down the intermediate state in the KHD formalism in a way that de-emphasizes (“direct”) or emphasizes (“indirect”) the role of the core hole. The intermediate state Hamiltonian is divided into that of the ground state Hamiltonian of the quantum system without a core hole, and a perturbative part that describes its change after the excitation step. The latter consists of the excitonic state created by the coupling of the core hole and the active electron and the interaction of this excitonic state with the other electrons in the system. The details of this division, which may be phrased in terms of Green’s function propagators, and the relative size of the two terms then facilitate the description of correlated materials [27]. The direct RIXS process emphasizes the role of the ground state term, and dynamical aspects of the core hole in the intermediate state are neglected. In practice, this description applies when the RIXS process is dominated by strong resonant excitations c → V ∗ from a core state c to unoccupied valence states V ∗ , followed by reversed transitions V → c from occupied valence states V back into the core hole. This situation is encountered in K-shell studies of low-Z molecules, where the 1s ↔ 2 p transitions involve molecular orbitals with large 2 p contributions. Similarly, L-shell (and M-shell) studies of the important 3d transition metal compounds are dominated by 2 p ↔ 3d (3 p ↔ 3d) transitions. In both cases, “up” and “down” transitions are strong due the rather localized nature of the valence states involved. Since low-Z K-edges and transition metal L-edges lie in the spectral range < 1000 eV, direct RIXS typically dominates in the soft x-ray region. The term indirect RIXS is used when the excitation process involves a “nonresonant” excitation to somewhat delocalized valence states that lie a few eV above the Fermi level E F . The origin of the name “indirect” is best seen from the example of K-shell RIXS spectra of 3d transition metal compounds, involving hard x-ray energies. If there were no localized 3d states around E F , the excitation of a 1s electron into delocalized 4 p states above the Fermi level would predominantly lead to a REXS process where the active excited electron itself refills the core hole. From a direct RIXS point of view, one would expect to see no inelastic losses. The presence of localized 3d states around E F , however, changes this picture. Then a RIXS signature

13.3 Quantum Formulation of REXS

641

may be observed which arises indirectly through an energy loss associated with an electron-hole excitation or “shake-up” within the d states. In this chapter we will restrict our discussion to the simpler direct RIXS process and illustrate the difference in its one-step and two-step description. The discussion of RIXS is facilitated by starting with the somewhat simpler REXS process.

13.3 Quantum Formulation of REXS The simplest case of (13.9) is resonant elastic x-ray scattering or REXS, defined as excitations from the electronic ground state through various intermediate core hole states back to the electronic ground state. Using the designation of electron states in the configuration model in Fig. 13.1, REXS consists of processes |b = |a. We consider only the resonant or second order term in the general KHD scattering expression (9.21) and ignore the elastic Thomson term and its possible interference with the resonant term. If we include the splitting of electronic states through nuclear vibrations, we have to be more specific in our definition of REXS. The initial state of a sample consists of an electronic ground state that in the Born–Oppenheimer approximation may contain several nuclear vibrational states. The true ground state may be prepared by cooling the sample. In contrast, the intermediate core excited states are “hot” so that higher vibrational states are always occupied. A true elastic scattering process that conserves energy in the photon system is a symmetric up and down process that leads back to the vibrational ground state. Processes that lead to higher vibrational final states or other low energy excited states are referred to as quasi-elastic. We will consider quasi-elastic processes as a specific case of resonant inelastic x-ray scattering or RIXS. In the following we shall first discuss the case of pure REXS, defined by a resonant scattering process from and back to a well-defined ground state.

13.3.1 The Fundamental REXS Cross Section For the REXS case we have |b = |a and with Ec − Ea = Eca , the general expression (13.9) reduces to

642

13 Quantum Theory of X-Ray Resonant Scattering

scat  dσa→a dσREXS 4π 2 ω1 ω2 αf2 4  = = R np2 k2 +1 d2 dω2 d2 dω2 λ22     a|ˆr ·  ∗ |cc|ˆr ·  p |a 2 1 p2   ×  δ(ω1 −ω2 ).(13.12)  c ω1 − Eca + i c /2 

We have deliberately maintained the δ-function notation that enforces energy conservation. As mentioned earlier, this has the important consequence that the intrinsic (i.e. in the absence of instrumental contributions) REXS linewidth of the scattered photons is not limited by the intermediate state width c . The multiplication with the δ-function in (13.12) effectively cuts out a slice of the intermediate state Lorentzian, resulting in an infinitely sharp peak. In practice, this corresponds to a measurement with δ-function-like instrumental bandwidths of the incident x-rays, determined by a monochromator, and of the scattered x-rays, determined by a spectrometer. The monochromator cuts out a narrow slice of the intermediate state Lorentzian, and this slice is then convoluted with the bandwidth of the emission spectrometer. This differs from the XES case, where the incident bandwidth does not enter and the emitted linewidth c is directly convoluted with the spectrometer resolution. For a δ-function-like spectrometer resolution, the measured XES linewidth will be c , since it is preserved upon convolution with a δ-function spectrometer width. Under the assumption of infinitesimally small instrumental bandwidths, the REXS (and RIXS) linewidth is not limited by the intermediate state width c as a consequence of strict energy conservation expressed by the Dirac δ-function. In contrast, the intrinsic linewidth in XES (and XAS) is determined by the natural decay width c of the intermediate (final) states.

13.3.2 REXS with Finite Instrumental Resolution In practice, the measured REXS cross section is obtained by considering the instrumental resolution functions of the monochromator that defines the incident bandwidth and of the emission spectrometer that analyzes the scattered photons. For gratingbased RIXS instruments, the resolution functions are taken to be Gaussians. The theoretical cross section must then be convoluted with the Gaussians. It turns out that for REXS, the δ-function in (13.12) can be replaced by a normalized Gaussian that contains the monochromator FWHM, 1 , and spectrometer FWHM 2 on equal footing with the following result.2

2

I would like to thank Faris Gel’mukhanov for the mathematical derivation of this result.

13.4 Spontaneous and Stimulated REXS Versus XAS

 dσREXS 4π 2 ω1 ω2 αf2 4  = R np2 k2 + 1 d2 dω2 λ22    a|ˆr ·  ∗ |cc|ˆr ·  p |a 2 1 p2   ×  G(ω1 −ω2 ),  c ω1 − Eca + i c /2 

643

(13.13)

where

2  a a (ω1 −ω2 )2 exp − G(ω1 −ω2 ) = √ . 2 ( 12 + 22 ) 2π 12 + 22

(13.14)

√ Here a = 2 ln 4 = 2.355 and the Gaussian is normalized to have unit integrated area (see Appendix A.2.2) like the δ-function it replaces in (13.12).3 The above formulation describes the case where the incident photon energy is defined by a monochromator of energy bandwidth (FWHM) 1 around ω1 and the scattered radiation is analyzed by a spectrometer with bandwidth (FWHM) 2 around ω2 . Hence there is only a REXS signal when the energy distributions between the incident and scattered photons overlap. In practice, the Gaussian of combined width G = 12 + 22 takes a slice out of the Lorentzian of width c so that the measured REXS linewidth is no longer determined by the intermediate state core hole width c but by that of the instrumental resolution function G .

13.4 Spontaneous and Stimulated REXS Versus XAS In our discussion above, we have assumed that both the incident and scattered radiation by atoms are analyzed in energy. If we do not analyze the scattered photon energy but only measure the energy integrated scattered intensity that falls into a detector of angular acceptance d2 around a certain scattering direction k2 , the double differential cross section (13.12) turns into a single differential cross section. The REXS intensity then depends only on the tuning of the incident energy, assumed to have a very small bandwidth, with (13.12) integrated over the emitted bandwidth dω2 . By use of the sifting property of the δ-function and denoting ω = ω1 = ω2 , we obtain  2    a|ˆr ·  ∗p2 |cc|ˆr ·  p1 |a  dσREXS 4π 2 (ω)2 αf2 4  = R np2 k2 +1   . (13.15)  c d2 λ2 ω − Eca + i c /2  This case is of interest since we can directly compare the REXS intensity to the XAS intensity, since both depend only on the tuning of the incident energy. 3

When for a given incident energy ω1 the Gaussian is integrated over ω2 , it reduces to unity.

644

13 Quantum Theory of X-Ray Resonant Scattering

For spontaneous REXS, we have n p2 k2 + 1 = 1 and the photons are omitted in random directions as a consequence of being created out of the random zero-point quantum vacuum. Their polarization vectors  ∗p2 lie in the plane perpendicular to the emission direction k2 into the detector with acceptance angle d2 . For stimulated REXS, we have n p2 k2 + 1 = n p2 k2 . Now the scattered photons are clones of the incident ones n p2 k2 that stimulate the decays, and the scattered photons propagate into the same direction k2 and with the same polarization p2 .

13.4.1 Spontaneous REXS Versus XAS We first consider spontaneous REXS and utilize the fact that the differential elastic scattering cross section is directly linked with the x-ray scattering length introduced in Sect. 6.2.3 by the relation (6.19) which in our case reads dσREXS = | f ( p1 ,  p2 )|2 . d2

(13.16)

From (13.15) we directly obtain the following expression for the spontaneous scattering length f ( p1 ,  p2 ) =

2π ω αf 2  a|ˆr ·  ∗p2 |cc|ˆr ·  p1 |a R . λ ω − Eca + i c /2 c

(13.17)

By assuming a single intermediate state |c of decay width = c , and denoting the resonance energy as E0 = Eca and the incident and scattered polarizations as  p =  p1 and  p =  p2 , we can write (13.17) in the form 2πω αf 2 f ( p ,  p ) = f  − i f  = R a|ˆr ·  ∗p |cc|ˆr ·  p |a λ   ω−E0 /2 × −i . (ω−E0 )2 + 2 /4 (ω−E0 )2 + 2 /4

(13.18)

For the resonant case, quantum theory directly provides the link between the real and imaginary parts of the scattering lengths4 f =

2(ω − E0 )  f .

(13.19)

The spontaneous REXS scattering cross section is given by This is the same as the classical relation between the corresponding scattering factors F  = f  /r0 and F  = f  /r0 given by (6.44). Note that in general, f  also contains a non-resonant Thomson contribution r0 Z according to (7.113). 4

13.4 Spontaneous and Stimulated REXS Versus XAS

645

spon ∗ dσREXS 4π 2 (ω)2 αf2 4 |a|ˆr ·  p |c|2 |c|ˆr ·  p |a|2 R = | f ( p ,  p )|2 = (13.20) d λ2 (ω − E0 )2 + 2 /4

which may be rewritten as spon

1 8π 2 ω αf 2 dσREXS R |a|ˆr ·  ∗p |c|2 = d 4π  λ2

 p

ca

×

2 /4 8π ω αf 2 . R |c|ˆr ·  p |a|2 (ω − E0 )2 + 2 /4 



(13.21)

p

σXAS =2λ f  p

The imaginary part of the forward scattering length f  p is linked to the x-ray absorpp tion cross section, σXAS = 2λ f  p by the optical theorem previously discussed in Sects. 7.6 and 7.7.2. The underbrackets correspond to our previous results given by (10.40) and (10.39). We can then write the spontaneous REXS cross section in the convenient short form spon

p

1 ca p σREXS = σ . d 4π XAS

(13.22)

Note the different polarizations p and p  . If we assume charge scattering and incident linear polarization, the integration of (13.21) over the solid angle of emission  will yield a factor of 8π/3 as for elastic Thomson scattering (see (6.21)), and we obtain p

spon

σREXS =

13.4.1.1

2 ca p σ . 3 XAS

(13.23)

Decomposition of Spontaneous REXS

According to (13.18), the complex scattering length has the real part f =

2πω αf 2 ω−E0 R a|ˆr ·  ∗p |cc|ˆr ·  p |a λ (ω−E0 )2 + 2 /4

(13.24)

which depends on the incident polarization and the polarization of the randomly scattered photons. It is zero at resonance, ω = E0 , and for charge scattering with linear polarization when  p ⊥  p (see (13.25) below). The imaginary part is linked by the optical theorem to forward scattering, and if we consider charge scattering, the polarization is preserved, p = p  . We then have

646

13 Quantum Theory of X-Ray Resonant Scattering

Fig. 13.3 Plot of the real and imaginary parts of the lineshape function defined by the curly brackets in (13.18), given at the top of the figure. The x-ray absorption cross section is proportional to Im L, while the resonant scattering cross section is proportional to L 2 . The squared real (red) and imaginary (light blue) parts in (b) have the same integrated value of π

Re L

L 4

Im L

(a)

= 1 eV 0=

3

0

2

L Re L Im L

2 1

Amplitude

0 -1 4

(b) 2

3

L 2 (Re L) 2 (Im L)

2 1 0 -4

f  =

-3

-2

-1

0 1 h (eV)

/2 2π ω αf 2 R |c|ˆr ·  p |a|2 . λ (ω−E0 )2 + 2 /4

2

3

4

(13.25)

The real and imaginary parts of the Lorentzian lineshape functions of f  and f are listed at the top of Fig. 13.3, and their lineshape contributions are plotted underneath in Fig. 13.3a. The absolute values squared of the real and imaginary parts of the lineshapes, which determine the scattered intensity, are illustrated in Fig. 13.3b. When we integrate the squared real (red) and imaginary (light blue) functions over photon energy, we find the same integrated value of π . The area under the total probability function, shown in black, therefore consists of equal probability contributions of the shown red (real) and blue (imaginary) parts to the total value of 2π. In accordance with the optical theorem, we may associate the imaginary part f  with the spontaneously forward scattered photons and the real part f  with photons scattered out of the forward direction. 

13.4 Spontaneous and Stimulated REXS Versus XAS

647

13.4.2 Stimulated REXS Versus XAS In addition to spontaneous forward scattered REXS there can also be stimulated REXS which naturally preserves the direction and polarization, k = k and p = p  of the photons driving the decays. The stimulated REXS cross section is obtained from (13.15) as     |c|ˆr ·  |a|2 2 stim 2 2 2 (ω) α 4π σREXS p   f R4 npk  = | f ( p )|2 =  (13.26)  c ω − Eca + i c /2  d λ2 For a single intermediate state the stimulated REXS cross section is related to the XAS cross section (10.39) and the two cross sections, written in the same notation, are linked according to stim npk 8π 2 ω αf 2 σREXS R |a|ˆr ·  p |c|2 = d 4π  λ2

 p

ca

×

2 /4 8π ω αf 2 , R |c|ˆr ·  p |a|2 (ω − E0 )2 + 2 /4 



(13.27)

p

σXAS p

p

where ca = ac is the dipole transition width previously given by (10.40). By combining (13.27) with (13.23) we can therefore write the total REXS cross section in the short form spon

σ σ stim 1 σREXS = REXS + REXS = d d d 4π



p

p

ca ca + npk

 p

σXAS .

(13.28)

This expression for the combined spontaneous and stimulated contributions has an important consequence, referred to as the “no-cloning” theorem [38–42]. While stimulated decays expressed by the second term in (13.28) indeed produce a cloned twin of the stimulating photon, the presence of spontaneous scattering, expressed by the first term, also has a forward scattering component. This spontaneous contribution prevents perfect cloning of a single photon. The theorem breaks down, however, when at large n pk stimulated scattering completely dominates.

13.4.3 Link to Semi-classical Results Our KHD results may be compared to the semi-classical Thomson, XAS, and REXS cross sections summarized in Sect. 6.4.1. The classical resonant XAS and REXS

648

13 Quantum Theory of X-Ray Resonant Scattering

cross sections are given by (6.46) and (6.47), which at resonance are given by E0 4π r0 E0 ( /2)2 spon , σREXS = σXAS . 2 2 (ω − E0 ) + ( /2) 3 λ (13.29) The corresponding KHD expressions for a given polarization are obtained from (10.38) and (13.21) as Classical: σXAS = 2λr0

Quantum: σXAS =

λ2 ac 2 ca ( /2)2 spon , σREXS = σXAS , π (ω−E0 )2 +( /2)2 3

(13.30)

where ac = ca is given by (10.40) or (13.21). This yields the quantum expression for the classical electron radius r0 =

ac λ . E0 2π

(13.31)

13.5 Intermediate State Interference Effects in REXS We have seen in Sect. 13.2.3 that a key question in the interpretation of experimental RIXS/REXS spectra is whether intermediate state interference effects may be neglected, which considerably simplifies the description. The role of such interference effects is best illustrated for the REXS case where the intermediate core hole state contains just two electronic or vibrational levels and REXS is described by (13.15). As discussed in Sect. 13.4, REXS is then directly related to XAS with the two intermediate REXS states corresponding to two final states in XAS. The XAS spectrum consists of just two peaks whose separation and relative intensities may be conveniently varied to explore their manifestations in the size of the REXS interference effects. In REXS, the initial and final states are the same well-defined ground state of the total nuclear-electronic system and the system can pass through the two energetically separated intermediate states, and the interference of these paths is of interest. The process is analogous to the quantum description of the double-slit experiment where photons from a source point may pass through spatially separated intermediate states, represented by the two slits, to the detector plane. In our treatment we assume that the two intermediate states decay with the same energy width back to the ground state. For our model calculations we assume a core to valence transition energy of about 1 keV, which according to Table 10.2 corresponds to a core hole lifetime width of about = 0.5 eV for both K- and Lshell excitations. In order to highlight the fundamental physics we assume incident radiation of negligible instrumental broadening and that the scattered intensity is measured at a certain momentum transfer with an energy integrating detector, as

13.5 Intermediate State Interference Effects in REXS

649

assumed in (13.15). Both REXS and XAS spectra are then only dependent on the tuning of the incident photon energy. The XAS spectra are calculated with the same value for the two peaks according to (10.55) and defining the resonant peak energies as E1 = Ec1 − Ea and E2 = Ec2 − Ea and the splitting energy = E2 − E1 . We write the XAS cross section (10.55) for two Lorentzian peaks in a convenient short form by defining the detuning energy as  = ω − E1 and lumping all prefactors in the cross section into the following intensity form IXAS = I1

2/4 2/4 + I , 2 2 + 2/4 ( − )2 + 2/4

(13.32)

where the Lorentzian lineshape function reduces to unity at resonance. By inspection of the XAS and REXS cross sections (10.55) and (13.15) we can rewrite the general REXS cross-section expression (13.15) which includes intermediate state interference in a corresponding intensity form with interference :

1step IREXS

 2   /2 /2   , = C  I1 + I2  + i /2  − + i /2 

(13.33)

where C is a constant scaling factor and the squared XAS intensities Ii arise from the fact that the REXS expression contains an angular transition matrix element that is the square of the XAS one if we ignore polarization effects. The corresponding REXS expression that does not include intermediate state interference effects reads  2step no interference : IREXS = C I12

 2/4 2/4 2 + I . 2 2 + 2/4 ( − )2 + 2/4 1step

(13.34) 2step

The size of the interference effect is given by the difference IREXS = IREXS − IREXS given by IREXS = C

⎧ ⎨

2I1 I2 

⎩

2/4 ( − ) +

2 2 /4 (− )+ 2 /4



⎫ ⎬ + 2 /4 ⎭

.

(13.35)

For any two-peak absorption spectrum IXAS , we can now calculate the corresponding one-step and two-step REXS cross sections and their difference IREXS . Since we know from (13.35) that the interference intensity scales linearly with the intensities, we pick I1 = 1 and I2 = 0.5. We then obtain the plots in Fig. 13.4 for the XAS and REXS intensities (scaled by setting C = 1) as a function of detuning energy  for four values of / . In Fig. 13.4a we have plotted the reference XAS spectrum according to (13.32) for four values of / as a function of detuning energy  relative to the first

XAS Intensity

(a)

1.2

XAS 0.8

= /3 = =3 =10

0.4 0.0

Interference intensity DirectREXS intensity

Fig. 13.4 a Reference XAS spectra according to (13.32) as a function of detuning energy , assuming I1 = 1 and I2 = 0.5 for four values of / indicated in color. b one-step REXS spectra including interference according to (13.33) with C = 1. c two-step REXS spectra according to (13.34) with C = 1. d REXS intensity difference of (b) minus (c) given by (13.35)

13 Quantum Theory of X-Ray Resonant Scattering

Total REXS intensity

650

(b) 1.5 REXS with interference

1.0 0.5 0.0 1.2

(c) REXS no interference

0.8 0.4 0.0

(d)

0.6

Difference = interference

0.4 0.2 0.0 -2

0 2 4 Detuning energy (eV)

6

XAS peak. The corresponding one-step REXS intensity (13.33) and two-step REXS intensity (13.34) are plotted in (b) and (c), and the interference intensity given by the difference of panels (b) and (c) and expressed by (13.35) is plotted in (d). We find that the interference contribution is constructive for < , while for > the interference changes from constructive to destructive near the XAS resonance positions. In particular, the expression identified by square brackets in (13.35) becomes 2 for = 0 and 2 for  = 0, revealing similar roles of the detuning energy  and the splitting energy , since they are both defined relative to the first peak energy E1 . The interference intensity thus falls off from its maximum value with a Lorentzian dependence according to

13.5 Intermediate State Interference Effects in REXS

IREXS = I1 I2

651

1 , 1 + 4x 2 / 2

(13.36)

where x =  or . Because of the long energy tail of the Lorentzian, the interference effects linger on to relatively large values of x/ . In REXS, interference of pathways through intermediate states of intrinsic decay width , separated by , modulates the elastic peak intensity as a function of the incident photon energy. The interference contribution is constructive for ≤ and for > changes from constructive to destructive close to the XAS resonance positions. Interference effects decrease with increasing values of and .

13.5.1 REXS Interference Contour Map In REXS/RIXS it is often convenient to view the intensity distribution in the form of a 2D contour map using the expression (13.12) with the axes given by ω1 and ω2 or with the ω2 axis replaced by the energy loss ω2 − ω1 . Such a contour plot for the REXS interference intensity is shown in Fig. 13.5 where we have replaced (13.35) by IREXS

2/4

a 2 (ω1 −ω2 )2  exp − = 2I1 I2  2 2 /4 2 G2 ( − ) + (− )+ + 2 /4 2 /4



(13.37) which explicitly distinguishes ω1 from ω2 linked through a Gaussian distribution of FWHM G according to (13.13). For the plots in Fig. 13.5 we have used I1 = 100, I2 = 50, = 0.5 eV, = 3 , and G = 0.25 eV. The line ω1 = ω2 where G = 0 correspond to the case shown in blue in Fig. 13.4d, with the intensity scaled by 100.

13.5.2 REXS Scattering Time Our treatment in the energy domain also allows us to picture the REXS process in the time domain, treated in more detail in [2, 12, 43, 44]. As discussed in Sect. 12.2.3, the scattering process at resonance may be described by a 1/e intermediate state lifetime given by (12.18) or

652

13 Quantum Theory of X-Ray Resonant Scattering

Fig. 13.5 REXS contour maps of the interference intensity IREXS given by (13.37) with I1 = 100, I2 = 50, = 0.5 eV, = 3 , and G = 0.25 eV, as a function of the incident photon energy ω1 and scattered energy ω2 (left) and energy loss ω1 − ω2 (right)

τ =

 .

(13.38)

When we detune the photon energy by  from the resonance energy E1 for a single intermediate state, the effective scattering time decreases according to τscat = √

 2

+ 42

.

(13.39)

For a detuning energy of  = 5 , the scattering time τscat reduces from the resonance duration τ by a factor of 10. For our value = 0.5 eV, for example, the resonance scattering time is τscat = τ 1.3 fs, and it reduces to τscat 130 as for a detuning of energy of  = 2.5 eV. The case τscat  τ has been referred to as the fast collision approximation [45, 46]. For two degenerate intermediate states we have = E2 − E1 = 0 and the scattering time also follows (13.39), where  is the detuning energy from the first resonance at E1 . When the interference intensity is measured at energy E1 , i.e.  = ω − E1 = 0, (13.39) is replaced by  τscat = √ . 2 + 4 2

(13.40)

Now the energy separation from the first resonance is equivalent to the detuning energy from the first resonance in (13.39).

13.5 Intermediate State Interference Effects in REXS

653

13.5.3 REXS Interference in Molecular Spectra: N2 and O2 We now illustrate the REXS formulalism for the resonant K-shell spectra of the N2 and O2 molecules. These cases are particularly pedagogical because the K-shell XAS spectra shown in Figs. 10.11d and 10.12c exhibit vibrational fine structure that in REXS (and RIXS) leads to intermediate state interference effects.

13.5.3.1

N2 REXS

We start with N2 studied in detail by Kjellsson et al. [32, 47]. In our discussion we follow the configurational transition diagram shown in Fig. 13.6 which follows our electronic state designation on the right side of Fig. 13.2 with |b = |a. The electronic ground state of N2 is that previously shown in Fig. 10.11a, b and at room temperature only the lowest vibrational state of energy Eν=0 = 0 is occupied owing to the relatively large vibrational splitting Eν=1 = 290 meV  kB T 25 meV. The XAS final state, which is the REXS intermediate state, is also vibrationally split as illustrated in Fig. 13.6, leading to the vibrational fine structure in the XAS spectrum. For N2 we have the situation 2 . The relative intensities in the XAS spectrum are calculated within the Born–Oppenheimer approximation [49] by modifying the electronic matrix elements by vibrational mode specific FranckCondon factors which depend on the overlap of the distance-dependent vibrational wavefunctions as shown for the XAS case in Fig. 10.11c. For the REXS case, we can take a shortcut and simply deduce the relative size of the squared matrix elements Mc = |c|ˆr · |a|2 , c = 0, 1, 2... from the relative intensities in the measured XAS spectrum. The theoretical XAS spectrum is then calculated according to (10.55) by use of = 120 meV for all vibrational components, with an intermediate vibrational state splitting of 230 meV, and transition matrix elements M0 = 1, M1 = 0.94, M2 = 0.49, M3 = 0.25, and M4 = 0.09, obtained by matching the relative intensities of the calculated spectrum to those in the experimental spectrum in Fig. 10.11d [48]. The so calculated XAS spectrum is shown in black on the upper left in Fig. 13.6a. This XAS parameterization was used to calculate the REXS spectra in Fig. 13.6b. The red REXS spectrum includes interference and was calculated with (13.13). The blue spectrum was calculated by neglect of interference through simplifying the double matrix elements in (13.13) according to (13.11). Their difference (red-blue), shown in Fig. 13.6c reveals the modulation caused by interference. The effect of interference is seen to be sizeable and as a function of energy may be constructive (red shading) or destructive (blue shading).

654

13 Quantum Theory of X-Ray Resonant Scattering

Configuration picture of N2 K-shell XAS and interference in REXS 402

~230meV

XAS

=4 =3 =2 =1

401

XAS final states = REXS intermediate states = core hole state |c + vibrational substates

=0

~120meV Rel. intensity

1.2

400

=2

electronic ground state |a in vibrational ground state

=1 =0

290meV

(b)

REXS interference no interference

(c)

difference = interference

1.0 0.8 0.6 0.4 0.2

REXS

XAS

Difference intensity

Incident energy (eV)

(a)

0.0 0.4 0.3 0.2 0.1 0.0 -0.1 -0.2

400

401 402 Incident energy (eV)

Fig. 13.6 a N2 XAS and REXS K-shell transitions in an electronic configuration picture that includes vibrational splittings in the electronic ground state |a and the core hole excited state |c labeled as in the configuration model in Fig. 13.2. REXS consists of transitions from the lowest vibrational state of the electronic ground state |a via all vibrational substates of the intermediate core hole state c back to the ground state. The theoretical XAS spectrum (black) is calculated according to (10.55), assuming the same core hole width = 120 meV for all vibrational peaks and vibrational final state splittings of 230 meV. The relative size of the transition matrix elements was simply matched to the intensities of the experimental spectrum in Fig. 10.11d [48]. b The red REXS spectrum includes intermediate state interference and was calculated with (13.13), while the blue spectrum neglects interference by simplifying the double matrix element according to (13.11). In both cases the same parameters were used. c Difference spectrum of the red minus the blue spectrum in (b). We have shaded the modulation induced by constructive and destructive interference in red and blue, respectively

13.5.3.2

O2 REXS

The corresponding theoretical XAS and REXS spectra of the O2 molecule are shown in Fig. 13.7. The relevant transitions have previously been shown in Fig. 10.12. The spectra in Fig. 13.7 were calculated using the same procedure as for N2 but using an electronic decay width of c = = 150 meV and a vibrational final (intermediate)

13.5 Intermediate State Interference Effects in REXS

Intensity

1.0 0.8

(a) O2 K-shell

655

XAS 125 meV

0.6 0.4

=150 meV

0.2

Rel. intensity

0.0 1.5

(b)

REXS interference no interference

1.0 0.5

Difference intensity

0.0 0.8 0.6

(c)

difference = interference

0.4 0.2 0.0 531 530 Incident energy (eV)

532

Fig. 13.7 Comparison of theoretical O2 XAS and REXS K-shell spectra. a XAS spectrum calculated according to (10.55) by use of the values in Fig. 10.12, c = = 150 meV, vibrational final state splittings of 125 meV and matrix elements obtained by matching the intensities of the calculated spectrum to those in the experimental spectrum in Fig. 10.12c [50]. b REXS spectra with interference, shown in red, calculated with (13.13) and without interference (blue) by simplifying the double matrix element (13.13) according to (13.11). We have used the same parameters as for the XAS spectrum. c Difference spectrum (red-blue) representing the modulation induced by interference which is constructive (red shading) at all energies

state splitting of 125 meV between the thirteen vibrational states indicated in the O2 XAS spectrum in Fig. 13.7a. The relative sizes of the thirteen matrix elements were deduced by matching the theoretical XAS spectrum shown in Fig. 13.7a to the experimental XAS spectrum in Fig. 10.12c.5 The REXS spectra calculated by including interference (red) and without interference (blue) are shown in Fig. 13.7b. In contrast to the N2 case their difference (red-blue) shown in (c) is positive at all energies, indicating constructive interference. Since for O2 is slightly less than , the interference intensity is always positive similar to the cases for ≤ shown in black and red in Fig. 13.4d. 5

Our synthesized spectrum in Fig. 13.7a agrees well with the high-resolution O2 XAS spectrum recorded by Strocov et al. [15].

656

13 Quantum Theory of X-Ray Resonant Scattering

13.6 Polarization and Spin Dependent Spontaneous REXS In practice, REXS is mostly used today for polarization dependent scattering and diffractive coherent imaging. In such experiments, the REXS intensity is typically measured as a function of the incident photon energy and polarization with no energy analysis of the scattered x-rays. The polarization of the scattered photons may also be analyzed [51–54]. We have seen that the spontaneous polarization dependent REXS cross section (13.15) is fully described by the polarization dependent scattering length given by (13.17). In the following we want to discuss a particularly elegant solution of f ( p1 ,  p2 ) for a magnetically aligned sample with isotropic charge distribution. The associated formalism was first developed by Hannon and Trammel in conjunction with Mössbauer scattering [46, 55] and later extended to its elegant modern form by Hannon, Trammel, Blume, and Gibbs which has been discussed in [45, 56– 60]. The quantum mechanical expression (13.17) for f ( p1 ,  p2 ) not only describes REXS but also XAS whose cross section is given by the imaginary part of the forward scattering length according to   σXAS = 2λ Im f ( p1 ,  p2 ), q = 0) .

(13.41)

This reflects the optical theorem as discussed in Sect. 7.6. For the XAS case, the REXS double matrix element simplifies according to  2 REXS : a|ˆr ·  ∗p2 |cc|ˆr ·  p1 |a =⇒ XAS : c|ˆr ·  p1 |a .

(13.42)

We have previously used the scattering length to phenomenologically describe the dichroic transmission through magnetic solids in Chap. 7 and dichroic diffraction in Chap. 8. All of our previous semi-classical derivations remain valid, but it is important to recognize the following key difference between the semi-empirical and quantum mechanical formulation. Our previous use of the photon energy and polarization dependent scattering length f p (ω) in Chaps. 7 and 8 was based on its empirical determination from experimental data. In this section we derive it quantum mechanically, which allows its calculation from first principles. The link between the scattering length f p and optical parameters β p and δ p given in Sect. 7.11.1 remains the same, so that our quantum derivation of f p may also be viewed as the quantum mechanical calculation of the energy and polarization dependent optical parameters.

13.6 Polarization and Spin Dependent Spontaneous REXS

657 scattered unit polarization

’ =( ’x , ’y , ’z )

incident unit polarization

=( x ,

y

, z)

y magnetization direction m z

x

k’

q

k

k planar sample

Fig. 13.8 Assumed geometry for the derivation of the polarization dependent x-ray scattering length for a magnetic sample. The photons are incident with wavevector k, and their linear polarization is described in the (x, y, z) coordinate system of the sample by a unit vector  = (x ,  y , z ). The ˆ aligned along the unit sample in the form of a film is assumed to have its magnetization M = M m ˆ shown in red, with two possible orientations ±m. ˆ The scattered photons propagate along vector m k with momentum change q and their polarization in the frame of the sample is   = (x ,  y , z )

13.6.1 The Polarization Dependent Scattering Length In the following we shall derive an expression for f ( p1 ,  p2 ) that describes polarization dependent magnetic scattering and absorption effects associated with x-ray magnetic linear dichroism (XMLD) and x-ray magnetic circular dichroism (XMCD). We choose the geometry shown in Fig. 13.8 and assume linearly polarized incident x-rays, so that the unit polarization vectors are real. Our use of linear polarization vectors is for convenience only, and they may be converted into circular polarization vectors as discussed in Sect. 13.6.1.1 below. The following treatment is a simplified version of that in [45, 56–58] and covers the cases of linear and circular magnetic dichroism measured in transmission through a magnetic film, treated phenomenologically in Sect. 7.11. It also provides the basis for diffraction of a film, which is nothing but elastic scattering modified by interference effects as illustrated in Fig. 8.16. In our treatment we will follow the notation in Fig. 13.8. We assume linearly polarized incident x-rays, so that the unit polarization vectors are real (see Sect. 3.2.6) and their components may readily be expressed in the Cartesian coordinate system of the sample. With  =  p1 =  ∗p1 and   =  p2 =  ∗p2 , the scattering length (13.17) becomes f (,   ) =

2π ω αf 2  a|ˆr ·   |cc|ˆr · |a R . λ ω − Eca + i c /2 c

(13.43)

The photons are incident on our planar sample along their wavevector k, and their linear polarization vector in the (x, y, z) frame of the sample is given by  =

658

13 Quantum Theory of X-Ray Resonant Scattering

(x ,  y , z ). The sample in the form of a film is assumed to be magnetically aligned ˆ along the surface normal, corresponding to the case of perpendicular magnetic with m anisotropy. The ferromagnetic orientation is defined by the unit magnetization vector ˆ which is unidirectional and can point either along ±z. The scattered photons m propagating along k have experienced a momentum transfer q. Their polarization has the general form   = (x ,  y , z ). In our sample coordinate system, the incident and scattered unit polarization vectors are given by ⎛ ⎞ ⎛ ⎞ x x (13.44)  = ⎝  y ⎠ ,   = ⎝  y ⎠ , z z where x2 +  y2 + z2 = (x )2 + ( y )2 + (z )2 = 1. The unit position vector in the dipole operators  · rˆ and   · rˆ in (13.43) may be expressed in terms of spherical tensors ⎛

⎞

⎛

ex

⎜ ⎟ ⎜ ⎜ ⎟ ⎜ rˆ = ⎜ e y ⎟ = ⎜ ⎝ ⎠ ⎝ ez

√1 2 i √ 2

 ⎞ (1) − C1(1) C−1  ⎟ ⎟ (1) + C1(1) ⎟ . C−1 ⎠ (1) C0

(13.45)

This will allow the calculation of the angular part of the transition matrix elements according to Sect. 10.4.6. For elastic scattering the double matrix element needs to connect the ground state |a via an excited state |c back to the ground state |a, and by considering the angular (1) (1) |a = l  m l |C−1 |lm l  momenta of the states we see that the matrix element c|C−1 (1)  couples the states m l and m l = m l − 1, and the operator C1 couples m and m l = (1) (1) |cc|C∓1 |a and the m l + 1. Therefore only the matrix elements of the form a|C±1 (1) (1) matrix element a|C0 |cc|C0 |a that preserve m l in the up-down process are nonzero. The double matrix elements are rewritten keeping only the non-zero terms and (1) ∗ (1) ) |a = (−1)q a|C−q |c∗ use of the operator identities c|Cq(1) |a = (−1)q c|(C−q (1) (1) and by use of a|C−1 |cc|C−1 |a = a|C1(1) |cc|C1(1) |a which is readily verified by inspection of Fig. 10.6. After some algebra one obtains a|r ·   |cc|r · |a =

    1   (1) |a|2 + |c|C1(1) |a|2 x x +  y  y + z z |c|C−1 2 

   ·

+



z ez

z ez  

2|c|C0(1) |a|2 −

(1) |c|C−1 |a|2 −

|c|C1(1) |a|2



ˆ ˆ (  ·m)(· m)

    (1) − i x  y −  y x ez |c|C1(1) |a|2 − |c|C−1 |a|2 , 

 ˆ (  ×)·m

(13.46)

13.6 Polarization and Spin Dependent Spontaneous REXS

659

ˆ = (0, 0, ez ) in Fig. 13.8. Our result where we have used the unit vector expression m separates the polarization dependence, expressed in terms of the real linear polarization components from the associated transition matrix elements written in terms of spherical tensors. The non-vanishing polarization dependent angular matrix elements M L|Cq(1) |ml for transitions between s, p, and d states are given in Fig. 10.6. By inclusion of non-resonant Thomson charge scattering given by (12.36) and neglect of the much smaller Thomson spin scattering (see (6.23) and (9.11)), we obtain the full expression for the quantum mechanical scattering length in the compact form f (,   ) = r0 F 0 (q) (  · ) 

 non-resonant Thomson charge

+

π ω αf 2 R λ



 ˆ ˆ − i C (  × ) · m ˆ . + B (  · m)( · m)  





A (  · )  

resonant isotropic charge

uniaxial magnetic: XMLD

unidirectional magnetic: XMCD

(13.47) The absolute value squared | f (,   )|2 is the differential scattering cross section which is seen to contain interference between the Thomson and resonant terms. The prefactors A − C are resonant matrix elements given by A=

(1)  |c|C−1 |a|2 + |c|C1(1) |a|2 c

B=

ω − Eca + i c /2

(1)  2|c|C0(1) |a|2 − |c|C−1 |a|2 − |c|C1(1) |a|2

ω − Eca + i c /2

c

C=

(1)  |c|C1(1) |a|2 − |c|C−1 |a|2 c

ω − Eca + i c /2

.

(13.48)

(13.49)

(13.50)

The real and imaginary parts in the expressions A − C of the resonant scattering length are separated by use of 1 ω − Eca + i c /2

=

c2 /4 2 ω− E ca −i . 2 2 (ω−Eca ) + c /4 c (ω−Eca )2 + c2 /4  



 real part

imaginary part

(13.51)

660

13 Quantum Theory of X-Ray Resonant Scattering

In (13.47) we have indicated by underbrackets the meaning of the various terms. The first term describes non-resonant elastic Thomson scattering by the atomic charge. The polarization factor   ·  is that previously derived classically for scattering by a single electron or atom in Sect. 6.2 and illustrated in Fig. 6.2. The polarization is preserved in the forward direction. The other terms describe elastic resonant scattering. The second term (prefactor A) describes resonant charge scattering by a spherically symmetric charge distribution and is the resonant analogue of non-resonant Thompson scattering with the same polarization dependence. The third term with prefactor B represents a uniaxial magnetic effect since it scales ˆ It also exists if the unidirectional vector m ˆ in a ferromagnet with the square of m. is replaced by a uniaxial magnetic symmetry in an antiferromagnet. It corresponds to the XMLD effect in XAS and arises from a spin-orbit coupling induced charge distortion along the magnetic axis. The charge distortion is revealed by rewriting the matrix element in the quadrupolar form along z given by 3z 2 − r 2 according to (1) 2|c|C0(1) |a|2 − |c|C−1 |a|2 − |c|C1(1) |a|2 # $  (1) (1) (1) (1) 2 2 2 2 = 3|c|C0 |a| − |c|C0 |a| + |c|C−1 |a| + |c|C+1 |a| . (13.52) 

 c

  3z 2 isotropic, r 2

The last term in (13.47), identified by the prefactor C depends on the existence ˆ as present in a ferromagnet. Its sign of a unidirectional magnetization direction m, ˆ and corresponds to the XMCD effect in absorption. depends on the direction of m

13.6.1.1

Linear Versus Circular Polarization Vectors

The polarization dependence of the scattering process is completely determined by ˆ that form the following irreducible components in the three unit vectors ,   , and m Cartesian coordinates   ·  = εx εx + εy ε y + εz εz ˆ ˆ = εz ez εz ez (  · m)( · m) ˆ = (εx ε y − εy εx ) ez . (  × ) · m

(13.53)

The corresponding expressions in a circular basis system are more complicated as seen from the operator expressions for linear and circular light propagating along different Cartesian directions in Table 11.1. The general case simplifies for special ˆ  z. This is the XAS case geometries such as for forward scattering q = 0 and k  m discussed in Sect. 7.11, leading to the XMCD effect in a circular basis and a Faraday rotation in a linear basis. The expressions for x-ray transmission derived previously in Sects. 7.11.3 and 7.11.5 can be shown to follow from the above scattering formalism.

13.7 Spontaneous Versus Stimulated REXS by an Atomic Sheet

661

13.7 Spontaneous Versus Stimulated REXS by an Atomic Sheet The “in-beam” forward scattered fields have a well-defined phase relative to the incident fields according to Sect. 7.5.4, and the scattered polarization remains in the plane perpendicular to the propagation direction but may be rotated for spin dependent scattering. Forward scattering has a spontaneous phase shifted contribution (see Sect. 7.5.4) and a stimulated in-phase and polarization conserving contribution due to the fact that photons clone themselves in the stimulation process. The spontaneous versus stimulated response of individual atoms treated in Sect. 13.4 may be extended to that of a thin sheet, whose plane is oriented perpendicular to the direction of the incident photons. The sheet is envisioned to have a small enough thickness that the exponential Beer-Lambert attenuation law can be approximated to second order in the sheet thickness. The intensity transmitted through an atomic thin sheet of thickness and areal density Na /A = ρa , where ρa is the bulk atomic number density, is given up to second order by the semi-classical expression (7.70)  Itrans = I0 1 −

Na 2λ f   A

−

  Na N2 8π  + 2λ2 ( f  )2 a2 . (r0 Z + f  )2 +( f  )2 A  3

A 

1st order XAS loss probability

incoherent REXS loss probability

coh. REXS gain= 2nd ord. XAS prob.

(13.54) This expression is valid in quantum theory under the assumption that all transition probabilities are spontaneous without inclusion of stimulated contributions. It holds to second order in the sheet thickness or number of atoms Na . The incident intensity is given by the quantum expression (10.9) as I0 =

n pk ωc n 0 ωc = . V pk A

(13.55)

The out-of-beam REXS probability is taken as an incoherent superposition of the atomic contributions (∝ Na ), while the in-beam (forward) scattering contribution may be equivalently described as a second order XAS or coherent forward scattered REXS contribution (∝ Na2 ). For a thin sheet we can distinguish contributions to the transmitted intensity in terms of their order in the sheet thickness or number of atoms Na . In first order, the XAS loss is linear in and Na , and we also have a REXS loss contribution due to “incoherent” out-of-beam scattering contributions by the atoms. The designation “incoherent” reflects the classical picture that interference of the fields emitted by randomly positioned atoms averages out. In second order, we have the last term in (13.54), which may be interpreted either as a “coherent” (in phase) forward scattering REXS contribution or arising from expansion of the exponential

662

13 Quantum Theory of X-Ray Resonant Scattering

Beer-Lambert absorption law (7.32) to second order in the sheet thickness . With increasing sheet thickness the coherently forward scattered REXS component will dominate. Neglecting the incoherent REXS term, we may write (13.54) in our quantum mechanical notation as  Itrans = I0 1 −

1 N a λ2 x + A π 2   

1st order XAS loss probability

2  N a λ2 x . A π



(13.56)

coh. REXS gain= 2nd ord. XAS prob.

The single atom spontaneous response discussed in Sect. 13.4 is replaced by the response of the entire sheet of Na atoms contained in an illuminated lateral area A. spon Since the single atom cross section σXAS represents the fractional atomic area that corresponds to unit absorption probability, the absorption probability of all Na atoms in the illuminated area A simply scales linearly with the number of atoms as Na = PXAS

N a λ2 x Na spon Na . = 2λ f  = σ A π A A XAS

(13.57)

The spontaneous second order REXS contribution due to coherent forward scattering is equivalent to the second order XAS term. It scales with the number of atoms squared Na2 as Na PREXS

N2 1 = 2λ ( f ) a2 = A 2 2

 2

N a λ2 x A π

2 =

1 Na2  spon 2 σXAS . 2 A2

(13.58)

Its positive sign leads to a transmission increase which is small for spontaneous forward scattering due to the small size of f  10−4 nm (see Fig. 7.12) but may be enhanced through stimulation, not included in (13.54).

13.7.1 Forward Scattering by an Atomic Sheet Upon stimulation by n pk of energy ω within the bandwidth pk , linked to the incident intensity by (see (10.8)) n pk = I0

λ3 , c pk

the spontaneously transmitted intensity (13.56) becomes

(13.59)

13.8 Resonant Inelastic X-Ray Scattering: RIXS

663



2  1 N a λ2 x  N a λ2 x 1 + npk Itrans I0 1 − + A π 2 A π     N2 Na I0 1 − 2λ f 0 + 2 a2 λ2 ( f 0 )2 1 + npk A A    spon 2   1 spon I0 1 − ρa σXAS + ρa2 2 σXAS 1 + npk . 2 

(13.60)

The spontaneous part of this expression reflects expansion of the exponential spontaneous Beer-Lambert law (7.32) to second order in the sheet thickness . It is implicity contained in the areal density Na /A = ρa , where ρa is the bulk atomic number density (see Table 7.2). The stimulated part of (13.60) reduces to the spontaneous forward scattered REXS intensity for npk = 1, as expected. In practice, film thicknesses exceed the validity range of our expression (13.60), and in Chap. 15 we will extend the KHD perturbation treatment of a thin sheet to cover higher incident intensities and films of arbitrary thickness.

13.8 Resonant Inelastic X-Ray Scattering: RIXS We have previously derived the general expression (13.9) for the RIXS cross section in the dipole approximation. In our derivation we maintained the Dirac δ-function in the second order KHD formula to conserve energy according to δ[(Eb − Ea ) − (ω1 − ω2 )]. In contrast to the REXS case, however, the final RIXS state |b will have a finite decay width b . We then need to replace the δ-function by a Lorentzian of energy FWHM b .6 Like the δ-function, the Lorentzian has to be normalized to have unit integrated area and the differential RIXS cross section then has the following fundamental form. The differential RIXS cross section, corresponding to a negligible instrumental linewidth contribution, is given by  4π 2 ω1 ω2 αf2 4  dσRIXS = R np2 k2 +1 2 d2 dω2 λ2 2  ω 2 ≤ω1   b|ˆr ·  ∗p2 |cc|ˆr ·  p1 |a   ×  L(ω1 −ω2 −Eba ), 

  c ω1 − Eca + i c /2   b Raman term

(13.61) where Ei j = Ei −E j and

6

The Lorentzian represents the final density of states of dimension [1/energy] according to (10.36).

664

13 Quantum Theory of X-Ray Resonant Scattering

L(ω1 −ω2 −Eba ) =

b2 /4 2 π b (ω1 −ω2 −Eba )2 + b2 /4

(13.62)

is the final state Lorentzian of width b whose argument ω1 −ω2 −Eba links it to the Raman effect in optical spectroscopy. Since it multiplies the intermediate state squared matrix element containing the larger core hole width c , the Lorentzian L effectively takes a slice out of the intermediate state width c in a RIXS map (see Fig. 13.5) that projects the incident and emitted energies ω1 and ω2 onto each other. The RIXS spectral resolution is therefore determined by the narrower width b . The cross section (13.61) establishes the link between optical Stokes Raman spectroscopy and RIXS, which is seen to extend the capabilities of optical Raman spectroscopy into the x-ray region. RIXS has therefore also been called “resonant x-ray raman scattering” which provides access to electronic or spin dependent final states of energies that exceed typical vibrational energies. The remarkable fact that the RIXS linewidth is narrower than the core hole width is a consequence of the second order KHD expression, where all paths through intermediate states do not enter in energy conservation. Second order RIXS and first order XES fundamentally differ in the achievable spectral resolution since according to (12.15) the XES linewidth is the sum of the core hole and final state widths. RIXS extends optical Raman scattering. It is element specific and allows the study of higher energy electronic final states that cannot be reached by direct dipole transitions. RIXS has a significant advantage over XES in terms of its intrinsic spectral resolution since the much larger core hole width which dominates XES is effectively eliminated in RIXS.

13.8.1 Two-Step RIXS The two-step RIXS differential cross section is obtained from the general expression (13.61) for negligibly small instrumental bandwidths by taking the sum over intermediate states out of the squared matrix element so that  4π 2 ω1 ω2 αf2 4  dσRIXS = R np2 k2 +1 2 d2 dω2 λ2   |b|ˆr ·  ∗p |c|2 |c|ˆr ·  p1 |a|2 2 L(ω1 −ω2 −Eba ). (13.63) × 2 2 (ω − (E 1 c − Ea )) + c /4 c b

13.8 Resonant Inelastic X-Ray Scattering: RIXS

665

This can be rewritten in terms of the dipolar transition width introduced for XAS in (10.40) and XES in (12.17) in the three state RIXS form  λ2  dσRIXS = 13 np2 k2 +1 d2 dω2 2π c b ×

2 1 ( c /2)2 8π 2 ω1 αf 2   c|(ˆ r ·  R |a p 1 2 [ω1 − (Ec −Ea )]2 + ( c /2)2 λ21 

 c p

absorption: ac1

×

2 1 ( b /2)2 8π 2 ω2 αf 2  R b|(ˆr ·  ∗p2 )|c , (13.64) 2 b [ω2 − (Ec −Eb )]2 + ( b /2)2 λ2 

 p

emission: cb2

where the underbrackets define the dipolar transition energy widths of the excitation and de-excitation steps in close analogy to the expressions (10.40) and (12.17) for XAS and XES. The two-step RIXS cross section is seen to correspond to two consecutive first order absorption and emission processes.

13.8.1.1

The Integrated Double Differential RIXS Cross Section

The two-step double differential RIXS cross section (13.64) contains two decay lifetime widths, those of the core hole intermediate state c and final state b . We can eliminate the final state width by integrating over all emission energies ω2 and obtain  λ2  dσRIXS = 12 np2 k2 +1 d2 4π p1  ac ( c /2)2 × 2 2 2 c c (ω1 − (Ec −Ea )) + ( c /2) %  p ∞ 2 ( b /2)2 × cb2 dω2 2 π −∞ b (ω2 −[ω1 −(Eb −Ea )]) +( b /2)2 b 



(13.65)

=1

to obtain the compact expression p1 p2    ac cb ( c /2)2 λ2  dσRIXS = 12 np2 k2 +1 . (13.66) d2 4π c2 (ω1 − (Ec −Ea ))2 + ( c /2)2 c b

In a simple three state model and denoting the intermediate state width as c = = X + A (see (10.35)) we can regroup the terms and can link the two-step single differential RIXS with the XAS cross section according to

666

13 Quantum Theory of X-Ray Resonant Scattering p2 p  cb dσRIXS 1  λ21 ac1 ( /2)2 = . (13.67) 1 + npk d2 4π π (ω1 − (Ec −Ea ))2 + ( /2)2 

 p

1 σXAS

& p p If we average the emission rate over polarization we obtain X =  cb2  = 13 p2 cb2 which is the radiative XES width that determines the fluorescence yield Yf defined in Sect. 12.4, so that p1  X λ21 ac 1  ( /2)2 dσRIXS = , (13.68) 1 + npk d2 4π π (ω1 − (Ec −Ea ))2 + ( /2)2   

 Yf

p

1 σXAS

where X is given by (12.22) or X =

L(Ne + 1) 8π 2 ω αf 2 R . 2 λ 3(2L + 1)(2c + 1)

(13.69)

If we additionally integrate over the 4π solid angle of inelastic emission which is isotropic, we obtain the RIXS cross section σRIXS =

 p1 X  np2 k2 +1 σXAS .  

(13.70)

Yf

We can summarize our results as follows. When the spontaneous (np2 k2 = 0) RIXS cross section is integrated over emission energies and angles, it becomes the absorption cross section times the fluorescence yield, as required by energy conservation. In this case all intermediate state interference effects sum to zero and cancel.

13.9 RIXS with Finite Instrumental Resolution The RIXS cross sections (13.61) and (13.64) have an intrinsic linewidth (i.e. for negligible instrumental bandwidths) determined by that of the final electronic state b . Even with state-of-the-art instrumentation, measured RIXS spectra typically still contain instrumental bandwidth contributions. The measured linewidth is then given by a convolution of (13.61) with the Gaussian monochromator and spectrometer resolution functions. In contrast to REXS, however, the two functions do not enter equivalently and one needs to carry out the convolutions separately. Also unlike REXS,

13.9 RIXS with Finite Instrumental Resolution

667

where the convolution of the intrinsic linewidth is facilitated by the energy conserving δ-function in (13.12), the RIXS convolution includes not only the expression for the intermediate state matrix element but the final state Lorentzian L(ω1 − ω2 − Eba ) given by (13.62). In this case there is unfortunately no general analytical solution for the convoluted lineshapes which have the form of Voigt functions.

13.9.1 The Case of Small Final State Width In practice, the RIXS final state |b typically lies above the ground state |a only by a small energy of order Eb − Ea 1 eV, where the final state widths b are associated with excited valence electron or vibrational states. In these cases we have b  c and one may approximate the final state Lorentzian by a Dirac δ-function. This significantly facilitates the convolution with the Gaussian monochromator transmission function of FWHM mo and yields an analytical expression. In the convolution, the energy-dependent prefactor in the RIXS cross section (13.61) may be approximated by a constant C since ω1 ω2 and pulled out of the convolution integral so that we obtain

dσRIXS d2 dω2

2 %∞    b|ˆr ·  ∗p2 |cc|ˆr ·  p1 |a   =C     c x − Ecb + i c /2 mo −∞ b

2  a a (ω1 −x −Eba )2 × δ(x −ω2 ) √ exp − dx 2 2 mo 2π mo 2     b|ˆr ·  ∗p |cc|ˆr ·  p1 |a  2 =C  G(ω1 −ω2 −Eba ), 

  c ω2 − Ecb + i c /2   b dispersive 





non-dispersive “resonance” term

“Raman” term

(13.71) √ where we have written the Gaussian in terms of the FWHM mo , where a = 2 In 4 see Appendix (A.2.2) C

 4π 2 ω1 ω2 αf2 4  R np2 k2 +1 λ22

(13.72)

and

2  a a (ω1 −ω2 −Eba )2 exp − . G(ω1 −ω2 −Eba ) = √ 2 2 mo 2π mo

(13.73)

Hence for a small intrinsic final state width b  c , the intrinsic final state Lorentzian in (13.61) may be replaced by a Gaussian that accounts for the finite

668

13 Quantum Theory of X-Ray Resonant Scattering

incident FWHM mo defined by a monochromator. Note that the monochromator function multiplies the intermediate state expression rather than convolutes it. This was pointed out by Tulkki and Åberg [11] in their discussion of early RIXS experiments [61, 62].

13.9.2 Reduction of RIXS to XES In the limit of large incident bandwidth, the Gaussian monochromator function becomes a constant and the RIXS linewidth is determined by the intermediate state width c . Then the narrow linewidth advantage of RIXS over XES is lost. Furthermore, in the absence of intermediate state interference effects, the two-step RIXS process becomes equivalent to XES, as expected for broadband excitation.

13.10 Examples of RIXS Capabilities We finish this chapter by showing characteristic K-shell RIXS spectra for the N2 and O2 molecules and L-shell RIXS spectra for transition metal atoms which are examples of our theoretical formalism.

13.10.1 K-Shell RIXS of N2 and O2 The K-shell RIXS spectra of the N2 and O2 molecules are particularly beautiful because they exhibit vibrational fine structure of both ground and excited electronic states. In their description, one needs to take into account all interference paths through the vibrational substates of the intermediate core hole state |c. Typically, the two-step RIXS description does not properly account for the intensities in the RIXS spectra although the measured peak positions are approximately predicted.

13.10.1.1

N2 RIXS

In contrast to the REXS process for N2 , illustrated in Fig. 13.6, RIXS decays involve energy transfer to the electronic and vibrational degrees of freedom of the sample. The lowest energy losses involve higher vibrational states of the electronic ground state of energy Eν=0 = 0 shown in Fig. 13.6. In cases where these small energy losses remain unresolved, one speaks of quasi-elastic scattering. In addition, there are also RIXS transitions associated with decays to vibrational substates of excited electronic states |b. The REXS and RIXS processes for N2 are schematically illustrated in Fig. 13.9.

13.10 Examples of RIXS Capabilities

669

(a) Electronic configurations and transitions for N2 3*u 1*g 3g 1u

spectator

eV 402.0

~120meV

RIXS 1-1 u

h 1 401eV

h2

3*u 1*g 2p 3g 1u

b

|b

9 eV

|a 2s 2u 2g

2u 2g

RIXS REXS

|a

h 1

3*u 1*g 3g 1u

402.5

Intensity 1.0

=2

|a =1 =0 ground state

(d) RIXS map - incident vs emitted energy

(c) RIXS map - incident energy vs energy loss excited electronic state |b

|b = 3g-1

~290meV

REXS

1u 1g

1s 1u 1g

~150meV =2 =1 =0

2u 2g

1u 1g

403.0

-1

|c = 1u

400.5

3-1 g

3*u 1*g 3g 1u

h1

=2 =1 =0

401.0

allowed hole dipole transition

1u 1g

~235meV

401.5

c~0.15 eV

|c

2u 2g

Incident energy (eV)

(b) Electronic and vibrational diagram

ground electronic state |a

excited electronic state |b

ground electronic state |a

402.0

REXS

0.5

401.5 401.0

0

Intensity sum

400.5

12.5

vibrational RIXS

10.0

2.5 5.0 7.5 Energy loss (eV)

REXS

0

387.5 390.0

392.5

395.0 397.5 400.0 Emitted energy (eV)

402.5 405.0

Fig. 13.9 a RIXS K-shell electronic configuration diagram and transitions for N2 . The electronic states are those previously shown in Fig. 10.11 with the core hole decay channel added. The RIXS decay consists of the dipole allowed transition from the core hole 1σu−1 state |c to the valence hole 3σg−1 state |b, and the REXS decay leads back to the ground state |a. b Inclusion of the vibrational substates in the diagram in (a), with the XAS spectrum taken from [48, 50]. c Experimental N2 RIXS/REXS intensity map, incident energy versus energy loss, recorded with an instrumental Gaussian resolution of FWHM 150 meV [47]. The energy loss spectrum shown at the botton is the intensity map summed over incident energy. d Same as (c) as a function of emitted energy (L. Kjellsson, private commun.)

Figure 13.9a shows the electronic configurations and transitions for the REXS and RIXS processes in N2 . The initial intermediate and final states states are labeled according to the configuration diagram in Fig. 13.2. The intermediate state reached by the 1σu → 1πg∗ transition is the same as the XAS final state whose vibrational fine

670

13 Quantum Theory of X-Ray Resonant Scattering

structure was previously shown in Fig. 10.11b. The excitation produces core holes of 1σu symmetry. The core hole decay corresponds to the hole in the intermediate core state |c “bubbling up” into an occupied valence state |b, subject to the dipole and parity selection rules. Therefore after 1σu → 1πg∗ resonant excitation only inverse parity allowed REXS hole transition 1σu−1 → 1πg−1 and the RIXS hole transition 1σu−1 → 3σg−1 are allowed. In particular, decays leading to 1πu−1 and 2σu−1 valence hole states, present in the XES spectrum in Fig. 12.7c, are parity forbidden. In Fig. 13.9b we include the vibrational energy splittings of the electronic ground state |a, intermediate state |c, and final state which is |b in RIXS and |a in REXS [32, 47]. At room temperature, only the lowest ν = 0 vibrational state of the electronic ground state |a is occupied since the ν = 1 state has an energy of Eν=1 = 290 meV >> kB T 25 meV. In Fig. 13.9c we show the experimentally observed N2 RIXS/REXS intensity map of the transitions in (a) and (b), recorded with an instrumental Gaussian resolution of FWHM 150 meV [47]. The intensity map corresponds to the RIXS spectra plotted horizontally as a function of energy loss relative to the REXS peak and vertically as a function of excitation energy. As the incident energy is increased, quasi-elastic transitions to higher vibrational levels of the electronic ground state are increasingly observed, as expected. The RIXS spectra shown at the bottom of Fig. 13.9c are the sum of those recorded at different incident energies. In Fig. 13.9d we have replotted the RIXS spectra in (c) by changing the horizontal scale from energy loss to emitted photon energy. The role of intermediate state interference effects is illustrated in Fig. 13.10. In Fig. 13.10a we identify two different excitation energies shown as the green (400.75 eV) and brown (401.75 eV) lines superimposed on the XAS spectrum [48]. In (b) we show as black points and curves the experimental spectra recorded by Kjellsson et al. [32] at the two exciation energies identified in (a). Superimposed are calculated spectra by inclusion of intermediate states interference effects (red curves) and without (blue curves) [32]. In both cases, the inclusion of interference shows considerably better agreement.

13.10.1.2

O2 RIXS

In Fig. 13.11 we illustrate the corresponding K-shell RIXS case for O2 . As for N2 , we illustrate in Fig. 13.11a the RIXS configuration diagram for O2 . In (b) we show the K-shell RIXS spectra recorded with FWHM 50 meV instrumental resolution by Hennies et al. [63]. The quasi-elastic RIXS spectrum, extending to an energy loss of about 2 eV, beautifully shows the development of the vibrational series with increasing excitation energy. In comparison with the N2 case, no vibrational peaks are missing due to intermediate state interference effects. This is due to the fact that for O2 the vibrational splitting is slightly less than the natural decay width of the intermediate state, so

13.10 Examples of RIXS Capabilities Fig. 13.10 a K-shell N2 RIXS configuration diagram with two different excitation energies identified in the XAS spectrum [48] in green (400.75 eV) and brown (401.75 eV). b Experimental RIXS spectra (black points and curves) recorded for the green and brown excitation energies identified in (a), and their comparison to theoretical RIXS spectra, calculated by inclusion of interference (red curves) and without interference (blue curves) [32]

671

(a) N2 Electronic and vibrational diagram eV 402.0 401.5

’’=2 ’’=1 |c = 1 ’’=0

401.0 400.5

-1

u

RIXS ’=2 ’=1 ’=0

REXS

|b = 3

-1

g

=2

|a =1 ground state =0

(b) N2 REXS/RIXS spectra: experiment & theory ground state |a

excited state |b

REXS

Emission intensity

401.75 eV

experiment

theory with interference no interference

400.75 eV

12

11

10

8 3 9 Energy loss (eV)

2

1

0

that the interference intensity is always positive as previously shown for the REXS case in Fig. 13.7c. Since the O2 XAS spectrum resembles an asymmetric Gaussian of about 0.7 eV FWHM, the collapse of the vibrational fine structure upon detuning of the incident energy from the peak (blue spectrum) to before the peak (pink spectrum) may be readily explained in a time-dependent picture as a reduction of the RIXS scattering time upon detuning. The coherently excited nuclear wavefunction develops until the core hole decays. As discussed in Sect. 13.5.2, the effective intermediate state

672

13 Quantum Theory of X-Ray Resonant Scattering

(a)

(b) O2 REXS/RIXS configuration diagram

eV

|c

~150meV

RIXS |b

RIXS |c |a

530.75

531.5

“quasielastic”

~125meV -1

|c = 1

u

1

+1

g

530.5 530.0

h |b = 3

-1

g

1

+1

Emission intensity

531.0

530.55

530.35

g

~7.5eV REXS

=2 ~360meV

|a = ground state

=1 ~180meV =0 =

530.05

0

8

6 2 4 Energy loss (eV)

0

Fig. 13.11 a K-shell O2 RIXS configuration diagram. The intermediate states |c reached by the 1σu → 1πg∗ transitions are shown as the XAS spectrum [50], with four different excitation energies indicated in color. b High-resolution (FWHM 50 meV) experimental RIXS spectra [63] at the excitation energies indicated in color

duration is reduced upon detuning (the fast collision approximation), and the RIXS process samples only the nuclear motion close to the ground state [64]. Another important point is the extremely narrow width of the quasi-elastic vibrational peaks in Fig. 13.7b. The natural lifetime widths of the vibrational peaks is of order of 1 meV. The measured RIXS width is about 50 meV, limited only by the instrumental resolution and not the intermediate state width of = 150 meV. This is a nice demonstration that in contrast to XAS, in RIXS the fundamental spectral resolution is not determined by the intermediate state lifetime but rather by the lifetime of the final state. The RIXS spectrum of the excited electronic state |b = 3σg−1 1πg+1 is centered around a loss energy of 7.5 eV, and the vibrational structure is only partly visible in the 6.6–7.2 eV region because it is obscured by an underlying broad feature. It arises from the presence of another state in a multi-configuration picture that interacts with the 3σg−1 1πg+1 state creating the complicated RIXS band around 7.5 eV energy loss [65]. For details the reader is referred to [63, 65, 66].

13.10 Examples of RIXS Capabilities

673

13.10.2 L-edge RIXS of Transition Metal Oxides The low energy excitations of the transition metal oxides are of great importance because of the many intriguing properties of such systems due to electron-electron correlation effects [67], such as low temperature superconductivity and colossal magnetoresistance, i.e. the dramatical change of the electrical resistance of manganesebased perovskite oxides in the presence of a magnetic field [68]. Transition metal compounds also play important roles in chemistry and biochemistry due to their ability to change their oxidation state [69], with bonds ranging from ionic in lower oxidation states to covalent in higher oxidations states. As such they exhibit calalytic activity which underlies biochemical transformations in nature and which are utilized in many industrial processes. Below we shall discuss the use of RIXS to explore the low lying excitations in La2 CuO4 which for many years has served as a basis for the development of related low temperature superconductors.

13.10.2.1

Cu L3 RIXS of La2 CuO4

The electronic structure of La2 CuO4 or LCO has been of keen interest because its central role in high temperature superconductivity. In transition metal compounds, the 3d orbitals are split by ligand field effects. Relative to the band-like 3d states in metals discussed in the following section, the orbitals can be assumed to be localized. For La2 CuO4 , the 2 p3/2 → 3d excitation step consists of promotion of a core electron into the half empty dx 2 −y 2 orbital. The importance of RIXS measurements arises from the fact that transitions between the d valence orbitals, which reveal the ligand field splitting, are optically forbidden by the dipole selection rule but may be observed by the L3 (2 p3/2 core state) RIXS process because it proceeds via an intermediate core hole state as shown in Fig. 13.12. In the ground state, Cu is in the 3d 9 configuration with an electron missing in one spin state of the highest energy dx 2 −y 2 orbital. We have illustrated the ligand field splitting of the 3d shell in the electronic ground state |a of La2 CuO4 in the lower left corner of Fig. 13.12a and have indicated the two possible spin states by different colors. The RIXS process illustrated for spin-conserving transitions in Fig. 13.12a involves the shown electron/hole configurations. In the intermediate state the active electron remains in the dx 2 −y 2 orbital. If the active electron itself fills the core hole, referred to a participator decay [1, 22], the scattering process is elastic. In the RIXS process there are three possible final states that can be reached from the intermediate state. They are created by bubbling up of the hole from the core to three possible valence hole states, as shown on the right side of Fig. 13.12a. The three final states lie above the ground state by different energies Eb with corresponding lifetimes b . As shown in Fig. 13.12b, the Cu L3 XAS spectrum of La2 CuO4 is dominated by a pronounced resonance (black curve) near 933 eV due to a 2 p3/2 → 3dx 2 −y 2 core to

674

13 Quantum Theory of X-Ray Resonant Scattering

XAS and RIXS at L3 edge of La2CuO4 intermediate state - core hole

(a)

dx2-y2 d3z2-r2 dxy dxz dyz

c ~0.5

eV

|c

2p3/2

final states - valence holes

h

2.12 eV

b=

0.28eV 0.20 eV 0.28eV

dx2-y2 d3z2-r2 dxy dxz dyz |a

2p core

b=

1.7eV 1.8 eV

2

933 eV

ground state

3d valence

h

1=

b

dx2-y2 d3z2-r2 dxy dxz dyz

|b

b

2p3/2

2p3/2 1.0 (c) RIXS

0.8 0.6 0.4 0.2 0.0

La2-xSrxCuO4 x=0.30 x=0.15 x=0.00

dxz dyz 0.8 Relative Intensity

Intensity (arb. units)

1.0 (b) XAS

dxy

0.6 0.4

d3z2-r2

0.2

933 931 937 0.0 935 -3.0 Incident photon energy (eV)

-1.5 -2.0 -2.5 Energy loss (eV)

-1.0

Fig. 13.12 a RIXS configurations for Cu L3 excitation ( 932 eV) in La2 CuO4 [70]. Opposite spin states are shown in different colors, and we only show transitions that preserve the spin. There are three final states which lie above the ground state by the listed energies Eb and have the shown decay energy widths b . b XAS spectrum of La2−x Srx CuO4 (G. Ghiringhelli and N. B. Brooks, private communication) for different Sr doping. c RIXS spectrum revealing the three final state configurations in (a). The blue curves are Lorenzians at the positions Eb with associated FWHM b . The dashed red curves are the sum of the dashed blue curves, and the solid red curve is the dashed red curve convoluted with a Gaussian, representing the instrumental resolution of 130 meV FWHM

13.10 Examples of RIXS Capabilities

675

valence transition. As indicated, the resonance changes in shape upon replacing some of the La by Sr atoms (G. Ghiringhelli and N. B. Brooks, private communication). The XAS resonance is similar to the previously shown spectrum of La1.85 Sr 0.15 CuO4 in Fig. 7.18 [71] and that of Cu-phthalocyanine in Fig. 11.3b [72, 73]. The resulting RIXS spectrum is shown in Fig. 13.12c. It consists of three peaks whose positions and decay widths are obtained from fits with Lorentzians, shown in dashed blue. Their summed contributions are shown in dashed read. The final lineshape fits, shown in solid red were obtained by convolution of the dashed red curve with a Gaussian of 130 meV, representing the experimental resolution of the grating emission spectrometer.

13.10.2.2

Observation of Magnons and Their Wavevector Dispersion

For elastic charge scattering the phase of the scattered photons is shifted by 180◦ so that the scattered photons have exactly the opposite phase to the incident ones. For elastic magnetic scattering the phase is either rotated by +90◦ or −90◦ , depending on whether the incident wavevector k is parallel or antiparallel to the magnetization direction. For inelastic charge and magnetic scattering the polarization of the scattered photons is the same as for the respective elastic cases, but the scattering phase is random. Since the 2 p3/2 states are spin-orbit coupled, they support both spinconserving and non-conserving transitions, making RIXS spin dependent through polarization dependent selections rules. Spin-flip transitions are only allowed for certain orientations of the antiferromagnetic spin axis relative to the lattice [14]. The polarization of the scattered radiation  ∗p2 is determined by that of the incident photons  p1 according to  ∗p2 ·  p1 for charge and  ∗p2 ×  p1 for spin scattering [74]. One can therefore distinguish charge and spin dependent scattering through polarization dependent selections rules. If the core states are strongly spin-orbit coupled as for the 2 p shell, they can support a change of the photon angular momentum and inelastic spin-flip scattering [75, 76]. As an example of spin-related features in RIXS and their energy dispersion with momentum transfer we show in Fig. 13.13 the Cu L3 ( 932 eV) excitation spectrum of La2 CuO4 [77]. The spectra shown in (a) as a function of momentum transfer q (G. Ghiringhelli and N. B. Brooks, private communication) extend in energy to the region of the d − d excitations shown in Fig. 13.12. At lower energy loss, a magnon peak, labeled B, is also observed. Its energy position disperses with q . Figure 13.13b shows the enlarged magnon region for a particular q , and the complete dispersion curve of peak B is shown as light blue data points in (c). The RIXS results are seen to be in excellent agreement with the dispersion curve measured by inelastic neutron scattering, which is superimposed as a dashed magenta curve.

676

13 Quantum Theory of X-Ray Resonant Scattering

La2CuO4 T = 15K

magnon peak

B q|| = 2.5 q|| = 1.57

d-d 2.5

2.0

q|| = 0.63 1.5

1.0

0.5

0.0

Energy Loss (eV)

(b)

(c) Energy Loss (meV)

RIXS intensity (arb. u.)

(a)

T = 15K q|| = +1.89

B

peak B dispersion

||

Fig. 13.13 a Extended Cu L3 ( 932 eV) RIXS spectra of La2 CuO4 as a function of momentum transfer q [77] and G. Ghiringhelli and N. B. Brooks, private communication. The spectra show the higher energy d − d excitations of Fig. 13.12 and a lower energy magnon loss peak labeled B. b Enlarged region of the magnon loss region. c Energy dispersion of the magnon loss peak as a function of q along the (−π, 0) → (π, 0) direction in the Brillouin zone, shown as light blue data points. Superimposed as a dashed magenta curve is the dispersion measured with inelastic neutron scattering

13.10.3 L-Edge RIXS of Transition Metals There is a distinct difference between the RIXS response of transition metal ions in non-metallic compounds such as La2 CuO4 discussed above and that of the elementary 3d transition metals, whose RIXS spectra have been extensively studied [35, 78–81]. In the 3d transition metals, the XAS excitations from 2 p core to 3d valence band states have been schematically illustrated in Fig. 10.14 and the corresponding XES decays in Fig. 12.9a. Their link in a configurational model is shown in Fig. 13.2. In our discussion we assume that the 3d valence electrons of the transition metal are not magnetically aligned and their charge distribution is isotropic, which strictly holds for cubic lattice symmetry and approximately even for hexagonal metals. The polarization independent RIXS process may then be pictured as illustrated in

13.10 Examples of RIXS Capabilities

677

Resonant excitation and spontaneous decays in 3d metals (b) REXS

(a) XAS

(c) RAS

(d) RIXS

ed band

F

t e ~ fs

s-p band

h

h

1

h

1

2

~ as

E

~ fs

2p core

Fig. 13.14 Simple picture of excitation and de-excitation processes associated with RIXS at the L-edge of 3d transition metals. Ignoring magnetic alignment, the metal valence band consists of delocalized s − p and localized d states, whose density of states is schematically illustrated. a Resonant excitation (XAS) of a 2 p core electron to empty 3d valence states on the attosecond timescale. b Resonant elastic x-ray scattering (REXS), involving up and down motion of the “active electron”. The emission occurs on the core hole clock timescale τ = / of order of femtoseconds with a probability expressed by the fluorescence yield as illustrated in Fig. 12.4. c Resonant Auger spectroscopy (RAS) following non-radiative core hole decay on the femtosecond timescale τ . In metals, the excited electron delocalizes into the s − p bands on a timescale te τ . d Resonant inelastic x-ray scattering (RIXS) where the core hole is filled by radiative decays of electrons in the filled 3d band

Fig. 13.14. The figure distinguishes the densities of states of the more localized 3d valence band states and the more delocalized s − p states. In the excitation or XAS process, illustrated in Fig. 13.14a, a 2 p core electron is excited into empty 3d valence states on the attosecond timescale. In the REXS process, shown in Fig. 13.14b, the excited electron decays back into the core hole and energy is conserved between the up and down motion of the “active electron”, so that the energy contained in the incident photon is re-emitted. The spontaneous emission process is in random directions, while the stimulated one is directional. Although the radiative dipole process has lower probability corresponding to a decay width X ∼ 10−3 eV, the total decay width contains a considerably larger ghost contribution A due to the faster Auger process as discussed in Sect. 12.2.5 with the transition widths shown in Fig. 12.10. After the excitation process, the intermediate state consists of a core hole with the active electron excited into empty 3d states. The core hole is effectively screened by the active electron in the intermediate state. There is little difference in the intermediate state created by “excitonic” active electron/core hole screening and collective core hole screening by metallic electrons. One may therefore assume that the active electron is absent due to rapid delocalization with extended metal states, as shown by

678

13 Quantum Theory of X-Ray Resonant Scattering

the blue arrows in Fig. 13.14c, d, with its screening role replaced by collective metal states. The physical origin of this active electron delocalization is the hybridization (mixing) between the more localized d band states and the more itinerant or delocalized s − p band states [74]. Although the active electron is excited with a larger dipole transition probability into the empty 3d valence states [82], it will quickly lose its angular momentum character through hybridization. This picture has emerged from extensive studies of charge transfer dynamics in core hole systems ranging from metals to atoms and molecules bonded to metal surfaces [1, 22, 35, 78, 79, 81, 83]. In the configuration picture of Fig. 13.2, the delocalization process of the active electron occurs in the intermediate state and it may be viewed to precede the core hole decay. This results in a decoupling of the excitation and decay steps, often called “dephasing” in the intermediate state. The intermediate core hole state is then well described by neglecting the influence of the active electron and RIXS may be described by the two-step RIXS expression (13.64). In effect, the RIXS process in transition metals is very similar to the XES process that follows threshold excitation. The advantage of threshold or resonant excitation to empty states just above the Fermi level is that it reduces multi-electron shake-up structure, as illustrated for Cu metal in Fig. 12.9d. In a simple model, the empty and filled d band degeneracies are given by the number of holes and electrons in our polarization averaged dipolar XAS transition width (10.59) and the XES decay width (12.22). The two-step RIXS cross section is then given by (13.68) or  X 1  dσRIXS = σXAS . 1 + npk d2 4π  

(13.74)

Yf

Here the polarization averaged XAS cross section is given by p

σXAS =

λ21  ac1  ( /2)2 π (ω1 − (Ec −Ea ))2 + ( /2)2

(13.75)

and the transition width for absorption is p1  ac =

8π 2 ω1 αf 2 L Nh R 3(2L + 1) λ21

(13.76)

and for emission is given by (13.69). If the emitted photon energy is analyzed by a spectrometer, the RIXS spectrum will then be representative of the density of the filled 3d band, as discussed in Sect. 12.3.4. For metals, the RIXS process may simply be envisioned as resonant absorption followed by XES.

13.10 Examples of RIXS Capabilities

13.10.3.1

679

XAS, XES, REXS, and RIXS Cross Sections

We can now summarize resonant cross sections involving transitions between core states of angular momentum c and valences states of angular momentum L that are occupied by Ne electrons and Nh holes by the following polarization averaged cross sections and transition widths. In all cases we assume Lorentzian transition widths (FWHM) denoted by i . We note that in practice there is often a trade-off between the width of resonant transitions and their peak value. This is due to either superpositions of vibrational peaks as shown for the K-shell resonances of N2 and O2 in Figs. 10.11 and 10.12 or band structure broadening as shown in Fig. 10.17 for the L3 XAS resonance in Fe, Co, and Ni. Under the assumption of natural Lorentzian lineshapes, the XAS case is given by (10.60) or λ2 XAS ( /2)2 (13.77) σXAS = π (ω − E0 )2 +( /2)2 XAS =

8π 2 ω αf 2 L Nh R . λ2 3(2L + 1)

(13.78)

The XES case is given by (12.16) and (13.80) as σXES =

  λ2 XES ( /2)2 1 + n pk 2 2 π (ω − E0 ) + ( /2)

XES = X =

L Ne 8π 2 ω αf 2 R . 2 λ 3(2L + 1)(2c + 1)

(13.79)

(13.80)

The REXS scattering cross section is obtained from (13.28) as  σREXS =

  REXS + npk σXAS , 4π

(13.81)

where for charge scattering  = 8π/3 according to (13.23) and REXS = XAS .

(13.82)

Finally, the two-step RIXS case is described by the angle integrated expression (13.74) or σRIXS = and

 RIXS  npk + 1 σXAS

(13.83)

680

13 Quantum Theory of X-Ray Resonant Scattering

RIXS = XES = X =

L Ne 8π 2 ω2 αf 2 R . 3(2L + 1)(2c + 1) λ22

(13.84)

It is apparent that for a given c and L the cross sections and transition widths may differ by factors that involve the orbital degeneracies (not including spin) given by 2c + 1 and 2L + 1 and the valence shell population expressed by Ne electrons and Nh holes. For the 3d transition metals, Fe, Co, and Ni the number of valence holes Nh and electrons Ne are given in Table 10.5. In the following we shall discuss the Co case as an example.

13.10.3.2

L3 RIXS of Co Metal

For the Co L-edge, we have c = 1, L = 2 and Yf = 8 × 10−3 . The spontaneous XAS and XES cross sections happen to be the same because Nh = 2.5 and Ne = 7.5, so that σXAS = σXES  σRIXS = 8×10−3 σXAS .

(13.85)

The RIXS case for L3 resonant excitation in the 3d metals is illustrated in Fig. 13.15a in a simple band model for the occupied and unoccupied 3d states previously used in Figs. 10.14 and 12.9a. The model assumes that at a given energy relative to the Fermi level, E F , the band averaged DOS may be described by a mixture of all d orbitals. Also, in the 3d transition metals, the numbers of d electrons in filled states Ne and holes in empty states Nh are not integers [74], and only their sum is given by the total spin orbital degeneracy of 2(2L + 1) = 10. RIXS spectra of transition metals are best recorded by tuning the incident energy just above the L3 threshold, determined by the Fermi level. The lack of excess incident energy avoids multi-electron shake-up structure [80], and the resonant excitation process increases the cross section. The measured resonant XAS and the spontaneous RIXS spectra for the L3 excitation in Co metal are shown in Fig. 13.15b [79]. Both consist of resonance like peaks associated with the filled and empty DOS, as expected. Although we have shown the XAS and RIXS spectra normalized to the same intensity, the RIXS intensity is considerably lower. According to (13.70) and (13.85), the angle integrated emission cross section is reduced relative to the absorption cross section because of the small fluorescence yield of 10−2 . In addition, only about one of 10−4 − 105 emitted photons is scattered into the solid angle of a typical emission spectrometer [84, 85]. In practice, the relative XES intensity is therefore smaller by a factor of order 10−6 − 10−7 . We will see in Chap. 15 that by stimulating the emission at high incident intensity, the RIXS intensity may become comparable to the XAS intensity.

13.10 Examples of RIXS Capabilities

RIXS at L3 edge of 3d metals (a)

c ~0.5

|c

eV

3d

delocalization

2p3/2

3d band model 3d holes DOS

k

3d electrons

h

h

1

2

b

3d

1

3d

|b

2p3/2

|a

2p3/2

Relative intensities

Fig. 13.15 a RIXS diagram for the L3 excitation in 3d transition metals, involving 2 p3/2 core and 3d valence electrons and holes. The intermediate state |c and final state |b are shown with the “active” electron in empty d states. In practice, it typically delocalizes as indicated by the blue arrow. b Measured spontaneous L3 -edge XAS spectrum for a Co/Pd multilayer [84] and RIXS spectrum for Co metal [79]. The Fermi energy is marked as E F

681

(b)

Co L3

0.8 0.6

spont. RIXS

spont. XAS

0.4 0.2 EF

0 770

772

774 776 778 780 Photon energy (eV)

782

784

786

13.10.4 RIXS of Chemisorbed Molecules: Polarization Dependence The properties of chemisorbed molecules on metal surfaces have been extensively investigated with many techniques because their behavior underlies the broad and important field of heterogeneous catalysis [22, 86, 87]. From a RIXS perspective, there is a significant difference in the description of free and chemisorbed molecules. We have seen in Sect. 13.8 for the fundamental cases of N2 and O2 that the intermediate state exhibits vibrational substates leading to interfering channels in the RIXS cross section. Also, in these cases one may assume that during the RIXS process the “active electron” remains as a spectator in the x-ray emission process [88]. As reviewed by Nilsson and Pettersson [22], the chemisorption bond mixes molecular and metal states so that the valence electronic structure of chemisorbed molecules on metal surfaces is in many aspects metallic. This causes a screening response of electronic excitations on the molecules due to coupling to the continuum of the delocalized metal states. After the resonant absorption step, the intermediate state consists of a molecular core hole with the active electron in an unfilled molecular orbital which already screens the core hole [89]. As discussed for the transition metal case above, the intermediate core hole may then be viewed as a fully screened state and may be represented as a single state in the RIXS process. The two-step RIXS

682

13 Quantum Theory of X-Ray Resonant Scattering

expression (13.64) may thus also be used for chemisorbed molecules. This simple picture has been supported by extensive experimental RIXS investigations pioneered by A. Nilsson in the late 1990s. It was also found that for weakly adsorbed species, the delocalization of the active electron may be slower so that the RIXS spectra may exhibit features reflecting “spectator” behavior of the active electron [22, 35, 83, 90, 91].

13.10.4.1

Polarization Dependence

Similar to polarization dependent XAS studies of chemisorbed molecules by use of the XNLD effect shown in Fig. 11.3, the RIXS intensity also strongly depends on the incident polarization. Studies of chemisorbed systems have not yet utilized analysis of the scattered polarization, but only the fact that the polarization vector  p2 necessarily lies in the plane perpendicular to the emission direction k2 because of the transverse nature of light. Analysis of the scattered polarization has been utilized in studies of molecular gases and correlated materials as discussed in Sect. 13.10.5 below. The dependence of RIXS on the incident polarization readily follows from the two-step RIXS description, which in the decay step restricts the dipole and parity allowed transitions back into the core hole. By simply detecting only photons emitted in symmetry chosen directions k2 , e.g. along the surface normal or the two in-plane directions, the emitted polarization  p2 is naturally restricted to the plane perpendicular to k2 [92]. This fact has been used extensively to determine the nature of the filled valence orbitals of oriented chemisorbed molecules [22]. An example was previously illustrated in Fig. 1.20 by polarization dependent RIXS spectra for benzene (C6 H6 ) molecules bonded to a Cu(110) and Ni(100) surfaces [93, 94]. The difference between the spectra for the two cases nicely shows how the weaker chemisorption bond to Cu leaves the benzene molecular orbitals largely unperturbed, while the stronger bond to Ni perturbs them significantly.

13.10.4.2

Example: RIXS of Glycine on Cu(110)

A particularly beautiful XAS and RIXS example is the case of the low symmetry molecule glycine, the simplest amino acid, chemisorbed on the low symmetry Cu(110) surface [95, 96]. The chemisorption geometry is shown in Fig. 13.16a, which was deduced from polarization dependent XAS spectra shown in (b) and RIXS spectra in (c). Glycine (NH2 CH2 COOH) consists of carboxyl (HC-OOH) and methylamine (H2 N-CH3 ) groups that bond together by replacing two C-H with a single C-C bond, as shown on the left of Fig. 13.16a. Upon chemisorption, the molecular skeleton is preserved, but the acidic H atom is removed and the resultant glycate molecule bonds to the Cu surface atoms as shown on the right side of the figure.

13.10 Examples of RIXS Capabilities

683

Glycate/Cu(110) X A S and X E S C HOMO C O

(a) LUMO

N [110] (z) [110] (y) [001] (x)

(b)

C K-edge

Int. (arb. units)

*

*

N K-edge

pz py

O K-edge

pz

pz

py

py

px

px

px C1 C2

Int. (arb. units)

400 410 420 530 540 550 Photon Energy (eV) Photon Energy (eV)

290 300 310 Photon Energy (eV)

(c)

Carboxylic Carbon Methylic Carbon pz pz

Nitrogen pz

Oxygen pz

py

py

py

py

px

px

px

px

20 15

10 5 0 15 10 Binding Energy (eV)

5

0 0 20 15 10 5 0 20 15 10 5 Binding Energy (eV) Binding Energy (eV)

Fig. 13.16 a Atomic model of glycine, showing the lowest unoccupied (LUMO) and highest occupied (HOMO) molecular orbitals and the geometry of glycate bonded to Cu atoms on the (110) surface. b Polarization dependent XAS spectra. The arrows indicate the excitation energies of the RIXS spectra. c Polarization dependent RIXS spectra measured for excitation of the C atom in the carboxyl-COO (labeled C1 in (b)) and in the methyl-CH2 group (C2 ), and for the N and O atoms excited at the arrow positions in (b) [95, 96]. The spectra are normalized to the same area, and the binding energy is referenced to the XPS 1s binding energy

The presence of C, N, and O atoms with directional covalent bonds provides an ideal case for demonstrating the power of polarized XAS and RIXS spectra. They provide atom specific and symmetry selective bonding information through spectra of unoccupied (XAS) and occupied (XES) orbitals. In all cases the electronic K-shell transitions involve spherically symmetric 1s core holes and orientation dependent 2 px , 2 p y , 2 pz valence orbitals. The polarization dependent XAS spectra shown in Fig. 13.16b were recorded with an instrumental resolution of 0.1 eV and provide direct orientational information

684

13 Quantum Theory of X-Ray Resonant Scattering

through the search light effect discussed in Sect.11.3.1. For glycine, the polarization dependence is most pronounced in the C and O XAS spectra. On both atoms there are π ∗ orbitals perpendicular to the bond axis and σ ∗ orbitals along the bond directions. The associated transition intensities are the largest when the E polarization vector is parallel to the spatial direction of the π ∗ and σ ∗ orbitals, respectively. The relative size of the resonances for E  x, y, z in coordinate system shown in (a) then directly reveals the orientation of the bonds between C, N, and O atoms. The polarization dependence of the RIXS spectra is shown in Fig. 13.16c. The spectra were recorded at incident energies marked by arrows in (b). The carboxyl carbon atom, C1 , was excited at its π ∗ resonance, while the methyl carbon, C2 , was excited at its C-H σ ∗ resonance [97]. The resolutions of the incident and scattered photons were set to about 0.5 eV. The spectra are normalized to the same area and put on a binding energy scale by subtracting the XPS 1s binding energy of the respective core levels. The valence electronic structure at binding energies less than about 6 eV corresponds to molecular orbitals responsible for the chemisorption bonds, while essentially pure intramolecular bonding orbitals give rise to RIXS features at higher binding energies. The detailed interpretation of the RIXS structure requires support through cluster models within the density functional framework [96]. It supports the chemisorption geometry shown in (a). The XAS and RIXS spectra in conjunction with detailed theoretical modeling reveal a stable glycate structure bonded to the surface through both oxygens and the nitrogen atom which allows the adsorbate to lie down on the surface. The bonding mechanism is found to be a combination of covalent and image charge contributions. The covalent part of the interaction mainly takes place via a hybridization of orbitals of z-symmetry, which offers the largest overlap between the adsorbate and the substrate orbitals.

13.10.5 Utilization of the Scattered Polarization In the soft x-ray region, the small fluorescence yield (see Fig. 12.10) has impeded analysis of the emitted polarization since it decreases the detected RIXS signal by more than an order of magnitude. It has only been recently implemented and utilized for a selected study at the Co L3 resonance in a superconducting cuprate [98]. It clearly will be utilized more in the future for the study of correlated materials based on improved spectrometers [18]. At higher x-ray energies the fluorescence yield is larger and Bragg diffracting crystals can be used [99, 100]. This was utilized around 1990 for polarization analysis of RIXS spectra recorded at the Cl K-edge (∼ 2800 eV) of chlorine containing hydrocarbon gases by Lindle et al. [101, 102]. One may think that the random molecular orientation in gases does not show a polarization dependent RIXS response, but this is not so. The basic origin of its existence is quite simple once one understands the polarization dependent response of oriented molecules as discussed in Sect. 13.10.4.

13.11 RIXS and Reduced Linewidth XAS (HERFD)

685

If a linearly polarized beam induces resonant transitions in randomly oriented molecules, the excitation probability of a molecule at a given incident energy will depend and its orientation relative to the incident E-vector. This is most easily understood by considering transition to unoccupied π ∗ or σ ∗ orbitals. For example, in diatomic molecules like N2 or O2 the π orbitals are perpendicular to the internuclear axis and the σ orbitals parallel to it. In ring-like molecules like benzene, the π orbitals are perpendicular to the ring plane and the σ orbitals are parallel to it. If the incident photon energy is chosen to correspond to a 1s → π ∗ transition, the E-direction will preferentially create core holes on molecules that are oriented with their π system along E. The emitted RIXS intensity will therefore be dominated by emission from these oriented molecules. Hence the emitted radiation will not have random polarization but an emission direction dependent polarization, similar to that of oriented chemisorbed molecules. The polarization dependence has been worked out in detail by Gel’mukhanov [12, 103] and may be written as ' ( σRIXS (,   ) = σ0 1 + R(3 cos2 θ − 1) , θ = ∠(,   ) 

*2 1 )ˆ 3 dω · dˆ ω − 1 R= 5

(13.86)

where σ0 is the isotropic cross section and dˆ ω is the dipole moment of the XAS transition and dˆ ω that of the XES transition, which are determined by the orientation of the empty valence orbital in the up transition and filled orbital in the down transition.

13.11 RIXS and Reduced Linewidth XAS (HERFD) In our configurational model in Fig. 13.2, the upper electronic core hole state |c typically has a larger width than the lower final state |b. This is utilized in RIXS, where according to (13.61) and (13.62) the intrinsic linewidth is effectively determined by the width b of the lower valence hole state. This valuable property of RIXS may also be exploited to improve the energy resolution in XAS spectra by a technique that has been referred to as lifetime reduced or high energy resolution fluorescence detection XAS, or “HERFD” for short [28, 104]. XAS spectra may be recorded by scanning the incident photon energy and monitoring any secondary electron or photon signal generated in the process of filling the created core hole. These secondary signals are proportional to the probability that the core hole was created by x-ray absorption in the first place. In principle, one should integrate the emitted electron or photon signal over energy and angle, as given by (13.70), but it is usually sufficient to select a fraction of the signal as utilized in partial electron or fluorescence yield detection [105].

686

13 Quantum Theory of X-Ray Resonant Scattering

XAS spectra of width smaller than the natural core hole width were first recorded by Hämäläinen et al. [106] by use of a RIXS-like experimental arrangement. They tuned the incident energy across an absorption resonance and used a spectrometer to detect a slice of the x-ray emission signal that was narrower than the core hole life time width. Today, the HERFD technique is typically employed for absorption resonances in the hard x-ray region (around 10 keV or more) whose intrinsic core hole width is large (a few eV). In these cases, the deep core hole is filled by electrons from upper core shells rather directly from the valence shell. In a configuration picture the HERFD final state is then a hole in an outer core shell that has a smaller natural width than the original deep core hole. The HERFD process is thus nothing but a RIXS process, but in contrast to conventional RIXS where the final state is a valence hole, it is a core hole in HERFD. The HERFD technique is based on reformulating the two-step RIXS expression (13.64), which for negligible instrumental linewidths has the form p p1    ac cb2 λ2  dσRIXS = 13 np2 k2 +1 d2 dω2 2π c2 b c b 

 determines resonance “intensity”

×

( b /2)2 ( c /2)2 . (ω1 − Eca )2 + ( c /2)2 (ω2 −Ecb ])2 +( b /2)2 



(13.87)

determines resonance width

As indicated by underbrackets, the resonant cross section consists of “intensity” and lineshape expressions. In the latter we typically have b < c , so that the multiplication of the two Lorentzians effectively takes a slice of width b out of the larger width c . The fact that the two Lorentzians are centered at different incident energies ω1 = Eca and emitted energies ω2 = Ecb is taken care of by a projection of the two energy axes onto each other as clarified by the following example of HERFD.

13.11.1 HERFD XAS at the Pt L3 -Edge As an example we discuss the sharpening of the L3 threshold resonance in Pt metal. This case has received considerable attention owing to the importance of Pt in catalytic and electro-catalytic processes [107]. Figure 13.17 illustrates the key energy levels and electronic transitions of importance and the RIXS configuration diagram of the excitation and emission processes. We have simplified the discussion by only considering the change of the threshold “white line” resonance which corresponds to transitions from 2 p3/2 core states to empty 5d5/2 valence states. In practice, the resonance is superimposed on an edgelike step [108], and the general RIXS case has been discussed by de Groot et al. [24].

13.11 RIXS and Reduced Linewidth XAS (HERFD)

(b) Electron/hole configurations

(a) Pt L3 energy level diagram

delocalization

binding energies

5d5/2

687

5d5/2 3d5/2

EF =0

2p3/2 c =5

|c

eV

c=11560eV

-2120 eV

3d5/2

L h

h

h

1

1

~11560 eV

h

2

1

|b

2

|a

~ 9440 eV

b =2

eV

a=0eV

5d5/2

5d5/2

9445

8 7 6

9440

5 4

9435

3 eff

2 1

9430 11550 11555 11560 11565 11570

(eV)

2

(d)

2125

-h

9

2130

2p3/2

1

10

(c)

Energy transfer h

Emitted energy h

2

(eV)

9450

3d5/2

-11560 eV 3d5/2 2p3/2

2p3/2

b=2120eV

2120 b

2115 c

2110 11550 11555 11560 11565 11570

Incident photon energy h

1

(eV)

Fig. 13.17 a Binding energy diagram of key energy levels of Pt involved in an L3 resonant transition from 2 p3/2 core states to empty 5d5/2 valence states. We have assumed that the 5d5/2 state is located at the Fermi energy with binding energy E5d5/2 = 0 eV, and the other binding energies are E3d5/2 = −2, 120 eV and E2 p3/2 = −11, 560 eV. b Electron/hole configuration diagrams relative to the ground state |a of zero energy. The intermediate core hole state |c of energy Ec = −E2 p3/2 is shown in the fast collision approximation with the active electron in an empty d state and a decay energy width c = 5 eV. The blue arrow reflects possible delocalization of the active electron before decay of the intermediate state. The final state |b of energy Eb = −E3d5/2 and decay width b = 2 eV lies above the ground state by the energy transfer ω1 − ω2 . c RIXS plane contour diagrams given by (13.87), representing the RIXS intensity distribution as a function of incident photon energy ω1 versus emitted photon energy ω2 . The thicker blue contour corresponds to the FWHM of the intensity distribution. Also indicated are the effective width FWHM eff = 1.86 eV given by (13.88) and the widths c and b . d Replacement of the emitted energy ω2 scale in (c) by the photon energy transfer ω1 − ω2

688

13 Quantum Theory of X-Ray Resonant Scattering

Emitted energy h

2

(eV)

9444 9442

FWHM contour

9440 9438

Normalized intensity

9436 1.0 h

0.8

2=

= 9440 eV

2

2+2eV

0.6

h

2

2-2eV

0.4 0.2

-

h

2

= XAS

0.0 11556 11558 11560 11562 11564 Incident photon energy h 1 (eV)

Fig. 13.18 Top: RIXS FWHM contour diagram according to (13.87) indicated by the rim of the light blue area. Bottom: Corresponding RIXS spectra measured with a flat-top detector window of different width, centered around the peak emission energy of 9440 eV. Black: infinitesimally small width, blue: 4 eV window width, red: very large width. The red curve represents the integration over all emission energies and is the conventional x-ray absorption resonance

In Fig. 13.17a we show the energy level diagram for the key Pt levels and the associated excitation (absorption) and emission transitions. We have used the binding energies E5d5/2 = 0 eV, E3d5/2 = −2, 120 eV, and E2 p3/2 = −11, 560 eV which are close to the exact values [109]. The corresponding RIXS diagram of the associated electron/hole configurations and the filling of the states is shown in Fig. 13.17b. The RIXS process involves the ground state |a of zero energy, the intermediate core hole state |c of energy Ec = 11, 560 eV and total decay energy width c = 5 eV, and the final state |b of energy Eb = 2, 120 eV and reduced decay width b = 2 eV. The assumed energy widths are close to the literature values [110]. In Fig. 13.17b the intermediate and final states still have the active electron as a spectator in the empty d orbital. As discussed above, we can assume for metals that the active electron will delocalize as indicated by a small blue arrow. The final 5d state then has a single hole in the occupied band and is the same as that in valence band photoemission. In Fig. 13.17c we show the RIXS intensity contour map according to (13.87) as a function if incident (ω1 ) and scattered (ω2 ) photon energies, and in (d) the scattered energy has been replaced by the energy loss ω1 − ω2 . For the simulations we have

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689 p

p

ignored polarization effects and set the transition matrix elements ac1 = cb2 = 1. The relative intensity is represented by colored contours in the figures, with the contour representing the FWHM of the intensity distribution shown as a thicker blue oval contour in (c). Its intersect with the blue horizontal emission energy line at ω2 = 9, 440 eV defines the minimum achievable energy width of eff = 1.86 eV, given approximately by + eff

1 . 1/ c2 + 1/ b2

(13.88)

This HERFD-XAS width is narrower that the conventional XAS width of c = 5 eV and is approximately equal to b = 2 eV illustrated in (c). The RIXS width expands into the XAS width by integrating the RIXS cross section over all emitted photon energies according to (13.70). The dependence of the HERFD-XAS width on the width of the detection window of the emitted photons, assumed to have a flat-top shape, is illustrated in more detail in Fig. 13.18. The point of the figure is to show how integration over an increasing window of the emitted intensity causes the HERFD-XAS spectrum to evolve into the conventional XAS one, as described by (13.70). We note that the reduction of RIXS to XAS spectra does not apply in general, but only for the two-step RIXS case where the excitation and emission processes can be decoupled [23, 25, 28, 104, 111].

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91. A. Nilsson et al., Appl. Phys. A 65, 147 (1997) 92. A. Nilsson, M. Weinelt, T. Wiell, P. Bennich, O. Karis, N. Wassdahl, J. Stöhr, M.G. Samant, Phys. Rev. Lett. 78, 2847 (1997) 93. M. Weinelt, N. Wassdahl, T. Wiell, O. Karis, J. Hasselström, P. Bennich, A. Nilsson, J. Stöhr, M. Samant, Phys. Rev. B 58, 7351 (1998) 94. L. Triguero, A. Föhlisch, P. Väterlein, J. Hasselström, M. Weinelt, L.G.M. Pettersson, Y. Luo, H. Ågren, A. Nilsson, J. Am. Chem. Soc. 122, 12 310 (2000) 95. J. Hasselström, O. Karis, M. Nyberg, L.G.M. Pettersson, M. Weinelt, N. Wassdahl, A. Nilsson, J. Phys. Chem. B 104, 11 480 (2000) 96. M. Nyberg, J. Hasselström, O. Karis, N. Wassdahl, M. Weinelt, A. Nilsson, L.G.M. Pettersson, J. Chem. Phys. 112, 5420 (2000) 97. J. Hasselström, O. Karis, M. Weinelt, N. Wassdahl, A. Nilsson, M.G. Samant, J. Stöhr, Surf. Sci. 407, 221 (1998) 98. L. Braicovich, M. Minola, G. Dellea, M.L. Tacon, M.M. Sala, C. Morawe, J.C. Peffen, R. Supruangnet, F. Yakhou, G. Ghiringhelli, N.B. Brookes, Rev. Sci. Instrum. 85, 115 104 (2014) 99. Z. Hussain, E. Umbach, D.A. Shirley, J. Stöhr, J. Feldhaus, Nucl. Instrum. Methods 195, 115 (1982) 100. S.H. Southworth, Nucl. Instrum. Methods 87, 247 (1994) 101. D.W. Lindle, P.L. Cowan, R.E. LaVilla, T. Jach, R.D. Deslattes, B. Karlin, J.A. Sheehy, T.J. Gil, P.W. Langhoff, Phys. Rev. Lett. 60, 1010 (1988) 102. D.W. Lindle, P.L. Cowan, T. Jach, R.E. LaVilla, R.D. Deslattes, R.C.C. Perera, Phys. Rev. A 43, 2353 (1991) 103. F. Gel’mukhanov, H. Ågren, Phys. Rev. A 49, 4378 (1994) 104. P. Glatzel, M. Sikora, M. Fernández-García, Eur. Phys. J. Spec. Top. 169, 207 (2009) 105. J. Stöhr, NEXAFS Spectroscopy (Springer, Heidelberg, 1992) 106. K. Hämäläinen, D.P. Siddons, J.B. Hastings, L.E. Berman, Phys. Rev. Lett. 67, 2850 (1991) 107. D. Friebel, H. Ogasawara, A. Nilsson, Operando x-ray and electron spectroscopy in oxygen electrocatalysis. in Surface and Interface Science, Vol. 8: Interfacial Electrochemistry, ed. by K. Wandelt (Wiley, New York, 2020) 108. B. Qi, I. Perez, P.H. Ansari, F. Lu, M. Croft, Phys. Rev. B 36, 2972 (1987) 109. A.C. Thompson et al., X-Ray Data Booklet, (3rd edn, LBNL, Berkeley, 2009). Available at http://xdb.lbl.gov 110. J.L. Campbell, T. Papp, At. Data Nucl. Data Tables 77(1), 1 (2001) 111. P.W. Loeffen, R.F. Pettifer, S. Müllender, M.A. van Veenendaal, J. Röhler, D.S. Sivia, Phys. Rev. B 54, 14 877 (1996)

Part IV

Multi-photon Interaction Processes

Chapter 14

Resonant Non-linear X-Ray Processes in Atoms

14.1 Introduction The Kramers-Heisenberg-Dirac (KHD) formalism has served the x-ray community well for the first 100+ years of x-ray science, and we shall see that it is valid even when the incident x-ray intensity is raised well above the level available at the brightest synchrotron radiation sources. The goal of this chapter is to explore when the KHD perturbation approach ceases to be a good description, and how the basic processes we refer to as x-ray absorption, x-ray scattering, and x-ray diffraction change in this largely unchartered x-ray territory that has been opened by the advent of XFELs. The development of a “strong-field” theory goes back to a seminal paper by I. I. Rabi in 1937 [1] where he considered the manipulation of nuclear spins by means of the magnetic field component of microwaves. Because of the low frequency of microwaves, the spins in a sample can still follow the magnetic field component of the EM wave. Also the long wavelength of microwaves, ranging from centimeters to meters, rendered the radiation naturally coherent so that its interaction with matter could be described by assuming classical EM waves of variable field strength. It was the development of powerful microwave sources during the Second World War for RADAR technology that furthermore paved the way for the development of nuclear magnetic resonance (NMR) in the 1940s. A key step in the development of NMR was the time-dependent description of the motion of the nuclear spin system in a strong EM field by Felix Bloch in 1946 [2], now known as the Bloch equations. In honor of the pioneering contributions of I. I. Rabi and F. Bloch to the development of the “strong-field” approach presented below, we shall refer to it as the Bloch-Rabi (BR) model. In optics, the theory is also referred to as the Maxwell-Bloch theory indicating its classical origin [3]. The powerful time-dependent BR model goes beyond the time-independent KHD perturbation treatment. At optical and higher frequencies the direct interaction of the magnetic fields of EM waves and the spins in a sample, treated by the original Bloch formalism, is negligibly small because the spins can no longer follow the highfrequency fields (see Sects. 7.3.3 and 7.3.4). Instead it is now the interaction of the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_14

695

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14 Resonant Non-linear X-Ray Processes in Atoms

electric field of the EM waves and the electronic charge in a sample that dominates. In analogy to the two-state (“up” and “down”) spin system, the electronic response of the sample can be cast into that of a two-state electronic system, as pointed out in 1957 by Feynman et al. [4]. With the invention of the laser in the 1960s, the BR formalism was readily extended, and the original spin-based Bloch equations were replaced by the “optical Bloch equations”, resulting in the birth of the entire new field of non-linear optics. Although the BR formalism has been extensively utilized in laser science, there was no need for x-ray scientists to use this formalism before the advent of XFELs since a simpler approach, the KHD perturbation theory, was available. The latter theory naturally lent itself to picturing x-ray interaction processes in a “photon” or particle picture, where a photon was lost in a sample through “absorption” or changed its direction through “scattering” or in a wave picture gave rise to “diffraction” phenomena. The need for a treatment beyond the KHD perturbation approach did not arise until the advent of x-ray free electron lasers in the new millennium. In this chapter we will discuss the BR theory in the modern language of the density matrix formalism for the case of a single atom. The atom is treated as a simple twolevel system, with electronic excitations of core electrons into valence holes, and de-excitations of valence electrons into core holes. The simple atomic case treated here nicely shows how the BR theory seamlessly evolves into the KHD theory at low x-ray intensities. The central goal of this chapter is to derive the equations governing the interplay of the populations of the lower and upper states of the resonantly driven two-level atom. The optical Bloch equations can only be solved analytically in certain cases, and these will be discussed and the solutions elucidated by use of parameters that are characteristic for the resonant response near x-ray absorption edges. We will also compare the time-dependent BR theory to the time-independent KHD theory, emphasizing the connection between the temporal properties of the incident radiation that enters into the BR theory and the energy bandwidth that enters into the KHD theory. Another important aspect is the relationship between stimulated forward scattering and absorption, which will be shown to satisfy a sum rule. We will also touch upon the famous Einstein rate equations, developed already in 1916/17 [5, 6], about 10 years before the development of quantum mechanics. As discussed earlier, Einstein’s theory for the first time distinguished spontaneous and stimulated emission, which was required to balance the energy in Planck’s famous blackbody radiation law formulated in 1900. We shall compare the results of the Einstein rate equation model, which assumes broad bandwidth radiation to the BR theory which specifically assumes narrow bandwidth radiation that is resonantly tuned. At the end of this chapter we will discuss additional fundamental aspects of the two-level atom which in the optical region is referred to as “resonance fluorescence” [7–10]. This topic considers the spontaneous fluorescence response of a two-level atom in the strong-field limit, measured at right angles to the incident beam and its linear polarization vector. We will see that the resonant florescence spectrum directly exhibits an energy splitting whose quantum electrodynamics treatment is of

14.2 X-Ray Induced Atomic Core to Valence Transitions

697

fundamental importance because it can be compared to the semi-classical Bloch-Rabi treatment. Resonance fluorescence has also played a role in identifying so-called antibunching behavior of photons [11, 12], in contrast to the bunching behavior discussed in Sect. 5.2.2. The atomic treatment presented here will be extended in Chap. 15 to solid samples, where the two-level atom is comprised of a collective multi-atom system.

14.2 X-Ray Induced Atomic Core to Valence Transitions While the description of the temporal behavior of a two-level electronic system at optical frequencies can be found in many textbooks, e.g. in [7, 9, 13, 14], we will consider the x-ray case where the electronic system is described by two levels in an atom, as previously illustrated for the KHD case in Fig. 10.3. Up and down transitions in such a “two-level” atom are induced by the electric component of the EM field or in a quantum picture by loss and generation of photons, and we consider the transitions and populations of a two-level atom illustrated in Fig. 14.1. In contrast to the KHD case, where the populations of the lower and upper states are assumed to be binary, either 0 or 1, our goal here is to formulate the transition probabilities between the two states in terms of time-dependent changes of the population distributions ρ11 (t) and ρ22 (t), which through the normalization condition ρ11 + ρ22 = 1 are temporally out of phase. In the x-ray case, the lower state is a core level and the upper state a valence level. We describe the core-to-valence dipolar transition width through a polarization average, which we define as equivalent “up”

Basic transitions in a two-level atom detuning energy energy

|b

relative population distribution

0

22

2

0

|a

h -

1

h

h

e-

11 + 22 =1

11

Fig. 14.1 Schematic of excitation and de-excitation processes in a two-level atom which in contrast to the KHD theory may have fractional occupation distributions ρ11 of the ground and ρ22 of the excited state, where ρ11 + ρ22 = 1. The incident x-ray energy ω is tuned close to the two-level resonance energy E2 − E1 = E0 where |ω − E0 | is referred to as the detuning energy

698

14 Resonant Non-linear X-Ray Processes in Atoms

and “down” transitions in accordance with (10.40) and (12.18) x =

8π 2 αf ω 8π 2 αf ω 2 = |b|r ·  |a| |a|r ·  p |b|2 . p λ2 λ2

(14.1)

For our two-level atom the transition width x is the same as the previously introduced phenomenological decay width  X shown in Figs. 10.3 and 12.10. In practice, the dipolar transition with x may be obtained from the peak value of the measured (spontaneous) absorption cross section σXAS linked by our previous expression (10.39) as x =

π σXAS . λ2

(14.2)

14.2.1 Interaction Energy and Hamiltonian: The Rabi Frequency Owing to the fact that the description of transitions between two spin or electronic states has a long history, the “two-level” case has been treated in great detail both classically and quantum mechanically [3, 15, 16]. The full quantum optics treatment of the coupling of a two-level atom to a single field mode is today known as the Jaynes-Cummings model [14, 16]. Before we present the treatment of the two-level case by the semi-classical optical Bloch equations, we need to briefly review the link of the classical field and quantum mechanical photon descriptions given in Chap. 3. This leads to a definition of the important Rabi frequency used in the BR formalism. Classically, the interaction energy Eint of a periodic electromagnetic wave with an atom is given by (14.3) Eint = −e r · E(r, t) = D · E(r, t), where D = −er is the dipole moment of the sample and the field is given by (3.13) or   (14.4) E(r, t) = E0 ei(k·r−ωt) + e−i(k·r−ωt) = 2 E0 cos(k · r − ωt). We note that for our field definition, the amplitude of the real wave field on the right side is given by 2E 0 . Following Rabi [1], one can describe the interaction of the field with a two-level atom in a time-dependent formalism that describes the frequency of the changes in the relative populations ρ11 and ρ22 . This characteristic frequency is called the Rabi frequency WR and is classically defined as [7, 14] WR =

D · E(r, t) D · E0 cos(k · r − ωt) = . 2 

(14.5)

14.2 X-Ray Induced Atomic Core to Valence Transitions

699

Quantum mechanically, one uses an interaction Hamiltonian that is equivalent to the classical expression Hint = −e r · E(r, t) = D · E(r, t),

(14.6)

where the operator E(r, t) is expressed by (3.61) and (3.68) as   E(r, t) = E0 a ei(k·r−ωt) + a† e−i(k·r−ωt) = E+ (r, t) + E− (r, t).

(14.7)

Here the operators a and E+ (r, t) destroy photons and a† and E− (r, t) create photons and  ωk . (14.8) E0 = 2 0 V The Rabi frequency may also be defined quantum mechanically within the timeindependent KHD perturbation theory. In this formalism the light-atom interaction is described by (13.1), which may be written in the dipolar form Wi→ f

2π = 

    f |D · E− |mm|D · E+ |i 2     ρ(E f ) δ(E f − Ei ),  m  Ei − Em

(14.9)

In the evaluation of (14.9) for a two-level atom, one makes the rotating wave approximation as discussed in Sect. 13.2. This corresponds to use of only half of Wi→ f , so that the quantum definition of the Rabi frequency in the KHD formalism is given by [7]1  Hint  Quantum Rabi frequency: WR = 2 (14.10) .  The importance of the Rabi frequency is that it is a fundamental building block in the optical Bloch equations which we will discuss below in Sect. 14.3.

14.2.2 The Rabi Frequency in the X-Ray Regime In the x-ray region, it is convenient to express the Rabi frequency by WR = R /. The energy width R , which we shall refer to as the Rabi energy, is then given by the quantum mechanical expectation value of the interaction Hamiltonian according to (14.10) or (14.11) R = WR = 2 Hint .

1

We follow Loudon’s notation of the optical Bloch equations and define the Rabi frequency (14.10) consistently with Loudon’s derivation in Sect. 2.4 yielding his (2.4.14) which leads to his (4.10.11).

700

14 Resonant Non-linear X-Ray Processes in Atoms

Its value depends on the incident intensity, and we shall see later in Sect. 14.11 that the situation where it becomes comparable with the total excited state decay width, i.e. R ∼ , is particularly interesting. The expectation value of the Hamiltonian  Hint  which describes the interaction of an electric field with a two-level atomic system, illustrated in Fig. 14.1, is evaluated as Hint  =  f | − er · E|i, where |i and | f  consist of products of photon and electron states as discussed in Sect. 10.2 for the resonant absorption case. We can therefore adopt the evaluation of the squared matrix element in that section and obtain in analogy to (10.3) | Hint |2 = | f |er · E|i|2 =

n pk c 2π 2 ω αf | b| r · |a|2 . (14.12)   

V pk 

coupling electron part  photon part

This gives the simple expression | Hint |2 =

n pk c λ2  x . V pk 4π  

(14.13)

0

It contains the photon flux 0 linked by (10.9) to the incident number of photons per mode, n pk , and to the incident intensity I0 as 0 =

n pk c n pk c pk I0 = 3 = , V pk λ ω ω

(14.14)

where V pk = λ3 ω/ pk according to (3.106) which reduces to V pk = λ3 for the case of unit bandwidth. The photon intensity I0 has the dimension [energy/(area× time]. The polarization averaged Rabi energy is obtained as2 R2 = 4 | Hint |2 =

n pk c λ2  x . V pk π

(14.15)

We obtain the following key relations between the key parameters entering into the KHD and BR descriptions of REXS.

2

This is consistent with Loudon’s (4.10.11) [7].

14.3 The Optical Bloch Equations

701

Coherent elastic scattering in the two-level scheme in Fig. 14.1 is defined by the incident radiation in terms of the number of incident photons n pk in the energy bandwidth pk or the associated intensity I0 and the atomic dipole transition width x . These quantities define the Rabi energy R according to R2 = I0

pk λ3 x , x = n pk 2 2π c 2π 2

(14.16)

where x / is the dipolar transition rate given by (14.1). This implies n pk = I0

λ3 . c pk

(14.17)

In the time-dependent BR theory the two-level response is determined by the Rabi rate R / which describes the buildup of the upper state population. This buildup is neglected in the KHD theory where the occupations of the lower and upper states are binary, either 0 or 1. In practice, the size of the non-linear electronic processes in Fig. 14.1 may be judged by the size of the spontaneous XAS cross section (see (10.55)) since it determines the initiation of the electronic processes. In the following will specifically denote it by a ♦ superscript as ♦ = σXAS

λ2  x (/2)2 , π  (ω − E0 )2 + (/2)2

(14.18)

where x is given by (14.1).

14.3 The Optical Bloch Equations We consider the transitions between two levels illustrated in Fig. 14.1 based on the following assumptions. • The two-level system consists of two electronic levels with energies E1 and E2 , separated by an energy E2 − E1 = E0 . After excitation the excited state decays with a natural rate /, where  is FWHM of the Lorentzian shaped energy distribution of the emitted radiation. • The energy bandwidth of the incident radiation may vary between the highly monochromatic limit   and the broadband limit = ω. In the monochromatic limit one can describe the resonant response as a function of “detuning” energy ω − E0 from exact resonance excitation.

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14 Resonant Non-linear X-Ray Processes in Atoms

14.3.1 Time-Dependent Transitions in a Two-Level System: Density Matrix Formulation We follow Fig. 14.1 and denote the fractional populations of the ground and excited states by real dimensionless quantities ρ11 and ρ22 , which are linked according to ρ22 = 1 − ρ11 . We also allow coupling between the two states, expressed as the complex electronic coherences ρ12 and ρ21 . They can be envisioned as the dynamic dipole moments induced by the oscillating electric field according to (14.3). The elements ρi j form a 2×2 atomic density matrix given by  ρ= We have

ρ11 ρ12 ρ21 ρ22

.

(14.19)

∗ Trρ = ρ11 + ρ22 = 1 and ρ12 = ρ21

and Tr(ρ)

2

= 1 for pure states . < 1 for mixed states

(14.20)

(14.21)

The last expression reflects the fact that the diagonal matrix elements are 0 or 1 for pure states but lie between 0 and 1 for mixed states. Let us consider a two-state electronic system described in the absence of an electromagnetic interaction by wavefunctions ψ1 and ψ2 of the two electronic states of energies E1 and E2 . In the presence of an interaction with a periodic electric field, the wavefunction of the system may be expressed as (r, t) = a(t) ψ1 (r, t) + b(t) ψ2 (r, t).

(14.22)

ρ11 (t) = |a(t)|2 and ρ22 (t) = |b(t)|2

(14.23)

Then the quantities

are just the probabilities of finding the system in the two states, respectively, and the coherences represent the interferences of the two states given by ∗ ρ12 (t) = a(t) b∗ (t) and ρ21 (t) = b(t) a ∗ (t) = ρ12 (t).

We have



ρ11 ρ12 ρ21 ρ22



 =

|a|2 ab∗ ba ∗ |b|2

(14.24)

.

(14.25)

Using the density matrix approach one can derive the important optical Bloch equations for a two-level-system by describing the time evolution of its density operator ρ(t) in the Schrödinger picture

14.3 The Optical Bloch Equations

703

ρ(t) = e−iHint t/ ρ(0) eiHint t/ ,

(14.26)

where Hint is time independent. The time evolution of the system is then entirely described by the density matrix according to the time-dependent Liouville-von Neumann equation i dρ(t) (14.27) = − [Hint , ρ(t)] . dt  The Rabi frequency enters through the relation (14.10) or Hint  =

 WR 2

(14.28)

The equations describe the time evolution of the coupled system. After making the rotating wave approximation which neglects terms of frequency much higher than the resonant frequency of the two-level system (see Sect. 13.2) and eliminating a phase factor by going to a rotating frame, we can state the Bloch equations as follows [7]. The optical Bloch equations describe the population changes in an upper electronic state, ρ22 , and lower state, ρ11 , separated by an energy E0 in response to an incident field of frequency ω under the condition ρ22 = 1 − ρ11 . Also included are interference contributions ρ12 and ρ21 between the states defined by (14.24). The incident photon energy ω may be slightly detuned by |ω − E0 | from exact resonance ω0 = E0 . The time-dependent process depends on the Rabi frequency WR , defined through the interaction Hamiltonian (14.28) and the total excited state spontaneous decay rate, /, which damps the oscillations of the populations. One may also include an additional dephasing rate deph /, which enters as  = + deph into the off-diagonal coherences. The coupled Bloch equations then read [7] dρ11 WR  dρ22 =− = −i (ρ12 − ρ21 ) − ρ22 dt dt 2 

(14.29)

and   ∗  dρ21 WR dρ12 = =i ρ12 . (ρ11 − ρ22 ) + i(ω0 − ω) − dt dt 2 2

(14.30)

Note that the rates / and  / enter differently by a factor of 2, which is discussed below. The first equation (14.29) expresses the time dependence of the populations of the ground and excited states which change with the Rabi frequency WR and spontaneous “down” transitions with a rate / that damp the oscillations through energy loss from the field. For resonant core-valence transitions in the x-ray regime,  is the total spontaneous decay rate of the excited atom in the absence of any external

704

14 Resonant Non-linear X-Ray Processes in Atoms

stimulus. Hence  =  X + A is the sum of the spontaneous radiative decay rate and the spontaneous Auger decay rate as discussed in Sect. 12.4. The last term ρ22 / in (14.29) causes a spontaneous decay of the population of the excited state. It may be thought of as a two-step process. First, in the excitation process energy is transferred from the field to the electronic system. However, the energy is not transferred back to the field but rather to a spontaneously emitted photon that is emitted into a random direction (hence mostly out of the beam) or to Auger electrons, with the Auger decay being 100 times more probable at soft x-ray energies. For a system of Na atoms, the populations are given by the fraction of atoms that are in the ground and excited states, respectively ρ11 =

N1 N2 and ρ22 = . Na Na

(14.31)

The second Bloch equation (14.30) expresses the time evolution of the coherences given by (14.24) which are products of the complex coefficients of the ground and excited state wavefunctions. From (14.30) we see that the wavefunction interference depends on the population difference driven by the Rabi rate, the energy mismatch determined by ω − ω0 , and a total damping constant  = + deph which includes the possible presence of a pure dephasing contribution deph , whose origin is discussed below.

14.3.2 Damping Constants: Longitudinal Versus Transverse Relaxation Dating back to the original use of the Bloch equations for the description of NMR, one often distinguishes two kinds of processes by the introduction of  and  in (14.29) and (14.30). In NMR, the two damping constants have specific names. In general, a spin may be oriented along any direction in space. If a static magnetic field is applied as in NMR, say along z, a spin will precess about the quantization axis z. The spin wavefunction for an arbitrary spin orientation is then described by a coherent superposition of the binary spin-up and spin-down basis states, where the coefficients of the two basis states are complex. Their magnitude reflects the tilt angle from the z field direction, and their orientation in the (x, y) plane perpendicular to the field is described by the phases of the complex coefficients [17]. One then distinguishes changes of the spin system along z, i.e. spin flips, and refers to this as a longitudinal process. In contrast, any change of the spin orientation in the (x, y) plane is referred to as a transverse process. Changes in the spin system are driven by an externally applied microwave field in the (x, y) plane, with the magnetic part of the EM field acting on the spin. Since in NMR the lower state is a pure spin-down basis state and the upper a pure spin-up basis state, a longitudinal process is a spin-flip process that changes the up and down populations. The characteristic time associated with population changes

14.4 Definition of Transition Rates in the BR Theory

705

is called the population or longitudinal relaxation time (or spin-flip time in NMR), T1 = / . In contrast, any change in the orientation of the spin projection in the (x, y) plane is just a change in their phase angle in that plane and is called the transverse relaxation or dephasing time (or spin dephasing time in NMR), T2 = 2/ deph . We have deph   2 1 2 1 + . (14.32) = , = , =  T1  T2  T1 T2 In laser science, the optical Bloch equations, also referred to as the Maxwell-Bloch equations, describe the change of the electronic population driven by the electric part of the EM field acting on the charge. One distinguishes two types of relaxation processes which affect the measured energy linewidth. Homogeneous damping processes preserve the central resonance energy of the atoms in the sample and the lineshape is Lorentzian. Inhomogeneous damping processes distribute the central resonance energy of the atoms in sample, and the lineshape is Gaussian. The most important homogeneous damping processes are that of spontaneous emission, leading to a natural Lorentzian lifetime width  of the excited state. Another process is due to elastic collisions in gases. From an atom point of view, the electron oscillation is interrupted but continues with a different phase after the collision. Such processes may be included through deph , also referred to as collision broadening [7]. In an equivalent picture one may also describe this process by a finite coherence time of the incident field, whose phase takes a random jump after the coherence time equals to the time between collision. If dephasing processes are absent, we simply have  =  in (14.30). Examples of inhomogeneous damping processes in gases are inelastic collisions that lead to a change in the excited state population or the Doppler effect which smears the central resonance frequency. In solids, important mechanisms are local inhomogeneities in the atomic environment or long range interactions between the valence electrons which either cause local variations in bonding energies or a smearing of the energy through band structure effects. In the soft x-ray region  is determined mostly by the Auger decay width. Measured soft x-ray resonances in solids are typically broader than this natural linewidth due to smearing of the valence states through bonding interactions or band structure effects and will be discussed in Sect. 14.4.1.

14.4 Definition of Transition Rates in the BR Theory The BR transition rates for the two-level atom are naturally defined in terms of the spontaneous transition matrix element, expressed through the dipole transition width x given by (14.1), the intensity-dependent Rabi transition width R expressed by

706

14 Resonant Non-linear X-Ray Processes in Atoms

(14.38), and the Bloch equation solutions for the populations ρ11 (t) and ρ22 (t). The BR transition rates are defined as follows. R2 ∞ ρ   11 A ∞ Spont. Auger: WA = ρ  22 x ∞ spon ρ Spont. Scattering: WREXS =  22 2 ∞ stim Stim. Scattering: WREXS = R ρ22 .  Absorption: WXAS =

(14.33) (14.34) (14.35) (14.36)

The populations are seen to be key parameters that determine the various transition rates, and in the following sections we will discuss in which cases analytical solutions for the populations can be obtained. The solutions will be illustrated by model calculations.

14.4.1 X-Ray Interaction Parameters for Model Calculations In the following sections we will take a look at the solutions of the optical Bloch equations by choosing realistic x-ray parameters. We have seen in Chap. 10 that strong resonances exist in atoms in the form of Rydberg resonances and in molecules in the form of transitions to unfilled molecular orbitals. In solids some of the strongest resonances are the 3d transition metal “white line” L-shell resonances shown in Fig. 10.16. In this chapter we choose our interaction parameters to reflect the Co metal L3 XAS resonance which has a peak cross section of σXAS = 6.25 Mb, previously illustrated in Fig. 10.17 and shown again in Fig. 14.2. Also shown in Fig. 14.2a is the considerably smaller spontaneous elastic scattering spon cross section, σREXS given by (13.23) which with the polarization averaged transition width (14.1) becomes3 spon

σREXS =

2  x λ2  x (/2)2 . 3  π  (ω − E0 )2 + (/2)2  

(14.37)

σXAS

The underbracket identifies the XAS cross section (14.18). The goal of the present chapter is to explore the increase of (14.37) through stimulation, as reflected by the arrow in Fig. 14.2a. We shall explore this through the additional term np2 k2 in the total KHD expression (13.28) and its link to the optical Bloch equations. 3

The factor of 2/3 arises from the emitted intensity node along the polarization vector.

14.4 Definition of Transition Rates in the BR Theory

Cross section (Mb)

1

(a) Co metal

absorption

L-edges

L3

L2

15

10 -1 stimulation

10

spont. elastic scattering

-4

10 -5 10 -6 750

5

theo XAS

770 780 790 800 Photon energy (eV)

810

820

Lorentzian =0.43eV exp XAS

Voigt V =1.6eV model

0 775

760

Co L3 absorption resonance

10

10 -2 10 -3

(b) Absorption cross section (Mb)

10

707

fit XAS

XAS

776

777 778 779 Photon energy (eV)

780

781

Fig. 14.2 a XAS cross section of the L-edge resonances in Co metal given by σXAS = 6.25 Mb and spon the spontaneous elastic scattering cross section σREXS = 3.5 × 10−3 Mb. In this chapter we will discuss the increase of the scattering cross section through stimulation, as reflected by the arrow. b Enlarged Co L3 resonance previously shown in Fig. 10.17b. We choose our modeling parameter for the BR treatment as indicated by the green curve of Lorentzian shape and FWHM  = 0.43 eV model = 6.25 Mb that matches the experimental value for Co metal. It and a peak cross section of σXAS corresponds to a transition matrix element defined by (14.1) of x = 0.34 meV

model To explore stimulation effects in REXS we shall use the model value σXAS = 6.25 Mb corresponding to x = 0.34 meV in conjunction with the natural Lorentzian linewidth  = 430 meV from Table 10.2. Our chosen value of x = 0.34 meV corresponds to the Lorentzian green curve in Fig. 14.2b, which has the same peak Co L3 absorption cross section as shown in (a), also shown as a black curve in (b). Our model value of the transition matrix element x = 0.34 meV is smaller than the value x = 0.96 meV (see Table 10.5) which corresponds to the shown blue Lorentzian curve of the same width  = 430 meV (also see Fig. 10.17b). We have chosen our model values to represent typical peak cross sections in the soft x-ray region, also reflected by the N2 and O2 XAS spectra shown in Figs. 10.11 and 10.12.

14.4.2 Practical Units and Beam Parameter Conversions Our model calculations employ specific units for the beam parameters. We will express the incident intensity I0 in units of [mJ/cm2 /fs]= [TW/cm2 ], typically used in x-ray laser science.4 The various physical quantities will be stated in the following units.

4

In the conversion of our units it is convenient to use 1 mJ/cm2 = 6.25×104 meV/nm2 . Other conversions are accomplished by use of Appendix A.1.

708

• • • •

14 Resonant Non-linear X-Ray Processes in Atoms

The incident intensity I0 is in units of [mJ/cm2 /fs]= [TW/cm2 ]. The photon energy is in [eV] and the wavelength λ in [nm]. All energy widths denoted i and pk are in [meV]. Cross sections are in [nm2 ], where 1 nm2 = 104 Mb.

With these units where I0 is measured in [mJ/cm2 /fs] and R and pk in [meV], the expressions for the Rabi energy R2 = I0

pk λ3 x x = n pk 2 2π c 2π 2

(14.38)

become for our Co L3 resonance model R [meV] = 3.8 n pk = 208.0



I0

λ3 I 0 I0 = 836.3 . pk pk

(14.39)

(14.40)

We will see in Sect. 14.6.3 that the mode-based description of the incident photons n pk given by (14.17) may be replaced by the number of photons, denoted n  , that are contained in the atomic core hole clock volume V under conservation of the photon density n pk /V pk = n  /V . In this case we have pk = 2π 2  and (14.40) is replaced with  in [meV] by λ3 I0 I0 = 42.4  

(14.41)

R2 = x n  = 42.4 x I0 .

(14.42)

n  = 10.55 and the Rabi energy is given by

14.5 Analytical Solutions of the Bloch Equations The key parameters determined by the Bloch equations are the temporal values of the lower state population ρ11 (t) and the corresponding upper state population ρ22 (t) = 1 − ρ11 (t). The absorption, emission, Auger, and elastic scattering rates are given by (14.33)–(14.36) in terms of the populations in conjunction with the dipolar energy widths x defined in (14.1), the atomic clock width  listed in Table 10.2,

14.5 Analytical Solutions of the Bloch Equations

709

and the Rabi energy width R expressed by (14.38) in terms of the incident number of photons or intensity. The Bloch equations have no general time-dependent solution and in general are solved numerically. However, a closed form solution for the populations ρ11 (t) and ρ22 (t) does exist for the following special cases. • Weak incident intensity. Here the BR theory gives similar results to the KHD perturbation theory under certain assumptions, discussed below. • Exact resonance excitation. This case corresponds to a small incident bandwidth or long coherence time of the incident photons, mimicking a long well-defined classical EM wavetrain. A time-dependent solution exists for any incident intensity. • Steady-state equilibrium. This situation occurs when the incident beam can act on the two-level atom for a sufficiently long time, t → ∞, that an equilibrium between the lower and upper state populations can be established.

We will discuss these three cases now.

14.5.1 Arbitrary Bandwidth: Low Incident Intensity The first case where an analytical solution for the excited state population can be found is when the incident intensity is low so that 4R < ,  (see (14.47) below). In this case the excited state population remains low as well. Under the initial conditions ρ22 (t = 0) = 0 and ρ12 (t = 0) = 0 one obtains [7, 18] by denoting the detuning energy E = E0 −ω and  = +deph ρ22 (t) =

(R /2)2    E 2 + ( /2)2      2(− /2) E 2 +( /2)2 t   exp − × 1+   E 2 + (− /2)2     [ (− /2) + E 2 ]  t E t  cos −  2 exp −  2  E + (− /2)2     2 E (− )  t E t  sin +  2 exp − .  2  E + (− /2)2 

For  =  , (14.43) reduces to

(14.43)

710

14 Resonant Non-linear X-Ray Processes in Atoms

0.008

(a)

=0

0.006 = h/

0.004

2

I0 = 100 mJ/cm /fs

0.002 22( t )

= 0.000 0.08

0

5

10

15

20

numerical solution

(b)

0.06

analytical expression

0.04

2

I0 = 1000 mJ/cm /fs =0

0.02 0.00 0

5

10 15 Time t (fs)

20

Fig. 14.3 a Dependence of upper state population ρ22 (t) on the time t according to the low intensity approximation given by √ (14.44) for two values of E = 0 and E =  = 430 meV. We have assumed R [meV] = 3.8 I0 (see (14.39)), with I0 = 1000 mJ/cm2 /fs,  = 430 meV and λ = 1.59 nm. The vertical black dashed line marks the core hole clock time τ . b Comparison of the low intensity analytical approximation (14.44) with the full numerical solution of the Bloch equations for an increased incident intensity of I0 = 1000 mJ/cm2 /fs. The onset of a discrepancy reveals the high intensity limit of the analytical solution (14.44)

ρ22 (t) =

      R2 /4 t E t 1 + exp − − 2 cos t exp − . (14.44) E 2 + 2 /4   2

The so-obtained excited state population is determined at a time t after the field is turned on instantaneously at t = 0. The off-diagonal density matrix elements or ∗ for ρ21 (0) = ρ12 (0) = 0 are given by coherences ρ12 = ρ21

  −iR /2 (i E +/2)t ρ21 (t) = 1 − exp − . i E +/2 

(14.45)

We have illustrated the dependence of the upper state population ρ22 (t) on the time t for our model parameters in Fig. 14.3. Figure 14.3a illustrates the excited state population ρ22 (t) given by (14.44) at an incident intensity of I0 = 100 mJ/cm2 /fs for our Co L3 resonance parameters specified in Sect. 14.4.1, for two values of the detuning energy E = 0 and E =  = 430 meV. As stated above, the analytical solution given by (14.44) is valid only if the incident intensity is not too strong so that the excited state population ρ22 remains relatively small. This is seen to be still the case for I0 = 100 mJ/cm2 /fs, where ρ22 (t) ≤ 8 × 10−3 . The analytical solutions shown in Fig. 14.3a are indistinguishable from the numerical solutions of the Bloch equations.

14.5 Analytical Solutions of the Bloch Equations

711

We also see that the black and blue curves for different detuning energies in Fig. 14.3a are identical for times that are shorter than the Co 2 p core hole clock time t < τ = 1.53 fs (see Fig. 12.4), marked as a vertical black dashed line. This is mathematically revealed by expanding the curly brackets in (14.44) up to second order in t, which yields  2  E + 2 /4 t 2 R2 /4 R2 t 2 = . ρ22 (t) = 2 E 2 + 2 /4   4 2 

(14.46)

g(t)

We see that at short times, ρ22 (t) is independent of the detuning energy E = E0 −ω. The validity range of (14.44) is demonstrated in Fig. 14.3b by comparing its predictions for a ten times higher incident intensity of I0 = 1000 mJ/cm2 /fs and E = 0, shown as a black curve, to that of the full numerical solution of the Bloch equations (red curve). Now the excited population has also increased by about a factor of ten and the full Bloch equation solution begins to deviate from the analytical solution (14.44). The analytical solution (14.44) is valuable since it is valid for all cases of incident bandwidth. It is therefore possible to directly compare it to the results obtained with other formulations such as KHD perturbation theory and the famous Einstein rate equations which we shall do in Sects. 14.6 and 14.9 below.

14.5.2 Exact Resonance: Arbitrary Incident Intensity Another case where an analytical solution of the Bloch equations exists is when highly monochromatic incident radiation is tuned to the resonance, i.e. E = 0. At low incident intensity, this is just a special case of (14.44), but at high incident intensity there is no analytical solution for finite detuning energy E that reduces to the special case E = 0. For E = 0 an analytical solution exists that is valid for all incident intensities at a time t after instantaneous turn on of the field at t = 0. In the absence of a dephasing contribution, i.e.  =  and defining the generalized Rabi frequency as ωR =

 R2 −  2 /16 

(14.47)

and under the initial conditions ρ21 (t = 0) = ρ12 (t = 0) = ρ22 (t = 0) = 0 the solution is given by [18] 

  R2 3 −3t/(4) sinh(ωR t) e 1 − cosh(ωR t) + . (14.48) ρ22 (t) = 2 4ωR  + 2R2

712

14 Resonant Non-linear X-Ray Processes in Atoms

(a)

0.30

4

numerical solution analytical, arb. I0 analytical, low I0

0.10

22( t )

2

I0 = 10 mJ/cm /fs =0

0.20

0.00 0

5 (b)

0.8

10

15 6

20 2

I0 = 10 mJ/cm /fs =0

0.6 0.4 0.2 0.0 0

5

10 15 Time t (fs)

20

Fig. 14.4 a Comparison of the time dependent upper state population ρ22 (t) for E = 0 and I0 = 104 mJ/cm2 /fs by use of the low intensity analytical expression (14.44) (black curve), the resonant arbitrary-intensity analytical expression (14.48) (light blue curve), and √ the numerical solution of the Bloch equations (dashed red). We have used R [meV] = 3.8 I0 , with  = 430 meV and λ = 1.59 nm (see (14.39)). b Same for I0 = 106 mJ/cm2 /fs. As previously seen in Fig. 14.3b the low intensity approximation (14.44) (black curve) increasingly breaks down with increasing I0 ≥ 103 mJ/cm2 /fs although it still exhibits the proper steady state (long time) value

Again we have ρ11 (t) = 1 − ρ22 (t). The excited state population ρ22 (t) approaches the equilibrium value (t → ∞) through a damped oscillation, indicated by the sinh and cosh terms as illustrated for our model parameters in Fig. 14.4. Expression (14.48) is valid for all incident intensities but is restricted to resonant excitation E = 0. It can be shown to reduce to (14.44) with E = 0 at low intensity. Equation (14.48) does not hold for broad band incident light as will be discussed in Sect. 14.9. Figure 14.4a illustrates the case for I0 = 104 mJ/cm2 /fs and (b) for I0 = 106 mJ/cm2 /fs for E = 0. Shown as black curves is the low intensity approximation (14.44) which according to Fig. 14.3b begins to break down around I0 = 103 mJ/cm2 /fs, and for the shown I0 values deviates significantly from the proper results, obtained either by the resonant analytical solution (14.48) (light blue curve) for arbitrary incident intensity or the numerical solution of the Bloch equations (dashed red). The increasing oscillations with time are the famous Rabi oscillations. The dependence of ρ22 (t) for resonant excitation, E = 0, expressed by (14.48) is illustrated in more detail over a broad range of intensities in Fig. 14.5. We see that initially the excited state population changes strongly with time and incident intensity. By use of a strong incident beam of large degeneracy parameter, one may create an ultrafast population inversion where the excited state population exceeds 0.5. As shown by the green curve for an incident intensity of 107 mJ/cm2 /fs, the

14.5 Analytical Solutions of the Bloch Equations

Population

22

(t)

Fig. 14.5 Excited state population as a function of time after instantaneous turn on of the intensity according to (14.48), for resonant excitation, E = 0, at various incident intensities I0 . We used √ R = 3.8 I0 [meV] and  = 430 meV

713

I0 [mJ/cm2/fs]

0.8

7

0.6

10 6 10

0.4

10

0.2

10 3 10

5 4

0.0

0

2

4 6 Time (fs)

8

10

Population

22

(t)

0.10 0.08 10

7

10

0.06

6

10

0.04

5

10

4

0.02 0.00

10

0

0.2

0.4 0.6 Time (fs)

0.8

3

1.0

population inversion can happen on a subfemtosecond timescale that is shorter than the atomic clock decay time τ = /  of order of a few femtosconds (see Fig. 12.4). When a multi-atom system is excited that way, the first spontaneous decays of some atoms can stimulate decays on other atoms which can amplify as in a laser, a process called amplified spontaneous emission or ASE. This was first demonstrated by use of XFEL pulses by Rohringer et al. [19]. The amplification process resembles the SASE process in an XFEL, with the difference that SASE starts from noise. As shown in Fig. 14.5a, the Rabi oscillations damp out in time and the excited state population stabilizes at an equilibrium value that depends on the incident intensity. We see that it takes about 5–10 fs at the highest intensities for the excited state population to reach a steady state. To create an excited state population of more than 10%, intensities higher than 103 mJ/cm2 /fs are needed. This means that the KHD perturbation theory which neglects population changes will break down. The relationship between the excited state population buildup as a function of time for different incident intensities and as a function of intensity for different times is shown in more detail in Fig. 14.6. The colored points on the vertical line at 1 fs in (a) correspond to the points of the same color in (b). It is instructive to consider (14.48) for the case of no damping, i.e.  = 0. We obtain ρ22 (t) =

1 [1 − cos(WR t)] , 2

(14.49)

so that the occupation of the excited state oscillates about the value 1/2 with the Rabi frequency WR . The occupation of the ground state undergoes the same oscillation with opposite phase since

714

14 Resonant Non-linear X-Ray Processes in Atoms 1

Excited state population

(a)

(b) After t = 1 fs

3

Intensity

0.8

10 4 10

mJ cm2 fs

10

0.6

10

5

x100

6

0.4 0.2 0 0

2

4 6 Time t (fs)

8

10 1

10

4

5

6

100 1000 10 10 10 Intensity (mJ/ cm2/ fs)

7

10

Fig. 14.6 a Excited state population as a function of time according to (14.48), for different incident intensities I0 . b Same as a function √ of intensity at different times. We have used parameters for the Co L3 -edge excitation, R = 3.8 I0 [meV] and  = 430 meV

ρ11 (t) = 1 − ρ22 (t) =

1 [1 + cos(WR t)] . 2

(14.50)

In retrospect, this justifies our earlier definition of the Rabi frequency in Sect. 14.2.

14.5.3 Excitations by Transform-Limited and SASE Pulses The analytical solution (14.48) gives the excited state population under the assumption that an incident coherently oscillating EM wave has acted on the electronic system for a time t and the population is probed at t. This corresponds to the assumption of a flat-top (FT) transform-limited incident pulse of length t. In practice, pulses typically do not have a FT shape and one needs to convolute the FT distribution with the detailed pulse shape. As discussed in Sect. 4.4.1 the pulse shape affects the Time-bandwidth product which, for example, differs by a factor of 2 for FT and Gaussian pulses (see Table 4.2). A more profound issue is that the internal structure of SASE XFEL pulses consists of coherent spikes of different widths and peak intensities [20]. In studies that depend on the temporal coherence, the different spikes act over different time periods t on the atoms causing dramatic spike-to-spike changes in the achieved upper state population (see Fig. 14.5). The action of SASE pulses then needs to be accounted for by a statistical average. One may assume a distribution of coherent spike lengths t during a pulse of length τp , so that the average upper state population is given by an integration of (14.48) over a range of coherence times t up to τ ≤ τp according to

14.5 Analytical Solutions of the Bloch Equations

715 I0

Population

22

0.8

[mJ/cm2/fs]

0.6 t

6

10

4

0.4

10

t

0.2 0.0 0

2

4

6

8

10

Time t or = t (fs) Fig. 14.7 Excited state population for two incident intensities I0 , distinguished by color, as a function of well-defined coherence time t (solid lines) after instantaneous pulse turn on according to (14.48) or averaged over a distribution of flat-top pulses of coherence time up to τ , expressed by √ (14.52) (dashed lines). We have used parameters for the Co L3 -edge excitation, R = 3.8 I0 [meV] and  = 430 meV

1 ρ22 (τ ) = τ

τ ρ22 (t) dt.

(14.51)

0

Assuming flat-top spikes of length up to τ , we obtain the analytical solution,    24ωR 1 − cos(ωR τ ) e−3τ/(4) R2 /4   1− ρ22 (τ ) = (/2)2 + R2 /2 τ ωR 16(ωR )2 + 9 2    16(ωR )2 − 9 2 sin(ωR τ ) e−3τ/(4)   − . (14.52) τ ωR 16(ωR )2 + 9 2 Figure 14.7 illustrates the expected behavior for two incident intensities. The integration over the pulse length (dashed curves) simply averages out the instantaneoustime Rabi oscillations and somewhat delays the onset of the population increase. For maximum coherence times much longer than the core hole lifetime, τ  τ = / , the term in wavy brackets in (14.52) approaches unity and ρ22 (τ ) → ρ22 (t). For large incident intensity, R   and long times t or τ we obtain the steady-state value ρ22 (∞) = ρ11 (∞) = 1/2.

14.5.4 Solution for the Steady-State or Long Time Limit We have seen in Fig. 14.4a that the excited state population for times exceeding the fundamental core hole clock decay time settles down to an equilibrium value. The long time solution may be obtained by solving simplified Bloch equations. As previously discussed in Sect. 14.5.1 we allow dephasing by use of  = +deph .

716

14 Resonant Non-linear X-Ray Processes in Atoms

The steady-state condition corresponds to setting the time derivatives of the density matrix elements to zero, and we obtain from (14.29) and (14.30)

and,

 WR  ∞  ∞ ∞ ρ22 = − i ρ12 − ρ21  2

(14.53)

   WR  ∞  ∞ = − (E0 − ω) + i ρ11 − ρ22 ρ∞, 2 2 12

(14.54)

where we have denoted the steady-state values of the density matrix elements as ρi j (t = ∞) = ρi∞j . We have written the equations with the population terms on the left and the coherences on the right. The density matrix solution in the steady-state limit t → ∞ for arbitrary detuning, ω − E0 , and in the presence of dephasing can be derived in closed form and is given by [7] ∞ (ω) = ρ22

R2  /(4) ∞ = 1 − ρ11 . (ω−E0 )2 + ( /2)2 + R2  /(2)

(14.55)

The solution is independent of the initial conditions. For the typical case  =  we obtain as follows. In the steady-state limit t → ∞ and arbitrary detuning, ω − E0 , the populations are given by ∞ (ω) = ρ22

(/2)2 R2   2 2  (ω−E0 ) +(/2)2 1+2R2 /  2

(14.56)

∞ ∞ ρ11 (ω) = 1 − ρ22 (ω).

(14.57)

and

The steady-state behavior expressed by (14.56) is illustrated in Fig. 14.8a for zero detuning ω = E0 , (black lines), and finite detuning, ω − E0 = /2 (blue lines), while in (b) we illustrate the change of R with intensity. Detuning is seen to delay the buildup of the upper state population, as expected. At the intersect R = , the upper state population in Fig. 14.8a becomes 1/3 and the lower state population is reduced to 2/3, as indicated by dashed lines. For larger values of R the populations equalize to ρ22 = ρ11 = 0.5. We will see that absorption and stimulated emission then become equal and the atoms become transparent.The gray shaded regions, corresponding to R  , indicate the validity of the KHD theory, which neglects the intensity dependence of the lower and upper state populations. The off-diagonal elements are obtained as

Equilibrium population

14.5 Analytical Solutions of the Bloch Equations

717

1.0

=0

(a) 0.8

= /2 =0.43eV

11

0.6 R

=

0.4 0.2

22

0.0

4

1

5

6

10 100 1000 10 10 10 2 Incident intensity I0 (mJ/cm /fs)

Linewidth (eV)

0.5 (b) 0.4 0.3

Validity range of KHD approximation

0.2

R

0.1 0.0

/ 10 1

10 100 1000 10 4 2 Incident intensity I0 (mJ/cm /fs)

∞ given by (14.56) and ρ ∞ = 1 − ρ ∞ for Fig. 14.8 a Steady-state solutions for the populations ρ22 11 22 resonant excitation E = ω − E0 = 0 (black) and off-resonance excitation E = /2 (blue), as a function of the incident √ intensity I0 . We have used the parameters for the Co L3 -edge excitation, R [eV] = 3.8 × 10−3 I0 and  = 0.43 eV. b Plot of the Rabi energy width R relative to the total decay width  as a function of I0 . The gray shaded regions identify the validity range of the KHD theory

∞ ∞ ∗ ρ21 (ω) = (ρ12 ) =

=

(R /2)(ω − E0 − i /2) (ω−E0 )2 + ( /2)2 + R2  /(2)

2 ∞ ρ (ω − E0 − i /2).  R 22

(14.58)

At low incident intensity, R  ,  , the last term in the denominator can be neglected and for  =  we obtain (14.45) in the limit t → ∞. At resonance we obtain the simple relation ∞ = −i ρ21

 ∞ ρ R 22

(14.59)

718

14 Resonant Non-linear X-Ray Processes in Atoms

(a) Change of peak population and width (b) Power broadening of distribution width 1.0

2

Incident intensity I0 (mJ/cm /fs)

0.6 0.5 0.5 22

0.4 upper 0.3

state

6 4

Rel. distribution width

=0.5eV R (eV) 5.0 1.0 0.5 0.25 0.05

11

Population

0.7

10

8

0.2

100

-2

0

2

-

105

106

1.03 1.02 1.01 1.00 100 1 10 1000 Incident intensity I0 (mJ/cm2/fs)

2

R=

=0.5eV

0 0.005

0.05

0.5 R

Detuning energy h

10

4

1.04

0.1 0.0 -4

10

3

1.05

22

lower 0.8 state

distribution width (eV)

11

and

0.9

5

(eV)

4 0

Fig. 14.9 a Plot of the lower and upper state population distribution functions (14.57) and (14.56) as a function of detuning energyω − E0 for different values of the Rabi energy R relative to the natural FWHM Lorentzian linewidth  = 0.5 eV. Note the complementary changes of the peak population values and distribution widths. b Distribution widths given by (14.60) as a function of√the Rabi energy R (bottom scale) or incident intensity (top scale) related by R [eV] = 3.8 × 10−3 I0 (see (14.39)). On the curve we have indicated as colored filled circles the widths of the curves of the same color in (a). The inset shows the relatively small width increase at lower incident intensities

14.5.5 Power Broadening of the BR Linewidth ∞ In Fig. 14.9a we show the change of the population distribution functions ρ22 (ω) ∞ and ρ11 (ω) given by (14.56) and (14.57) as a function of detuning energy ω−E0 for different values of the Rabi energy R . For convenience we have assumed a natural lifetime width  = 0.5 eV, instead of  = 0.43 eV in Fig. 14.8a. It is intuitively clear that the peak population values determine the peak transition rates and their energy distributions determine the detuning response. We will formally define the different transition rates and linewidths in terms of population distribution functions and transition matrix elements later, and at this stage the qualitative link suffices. ∞ ∞ and ρ11 given In Fig. 14.9b we plot the widths (FWHM) of the distributions ρ22 by

FW H M =



 2 + 2R2

(14.60)

as a function of the Rabi energy R (bottom scale) and the incident intensity √ (top scale). The scales are related in the shown units by R [eV] = 3.8 × 10−3 I0 (see (14.39)). The shown colored circles indicate the widths of the curves of the same

14.6 Link of KHD and Low Intensity BR Rates

719

color in (a). The distribution width increase with intensity is referred to as power broadening. The population distributions as a function of detuning energy in Fig. 14.9a for the lower and upper populations are mirror images around the average value 1/2. The peak population values follow the black curves in Fig. 14.8a, except for a slight difference due to the value  = 0.43 eV used in that figure and  = 0.5 eV used in Fig. 14.9a. At low incident intensity, R  , the distributions (14.56) and (14.57) assume the Lorentzian form, e.g. ∞ = ρ22

 2 /4 R2 .  2 (ω − E0 )2 +  2 /4

(14.61)

At high intensity 2R2   2 , the distributions become ∞ ρ22 =

R2 /4 (ω − E0 )2 + R2 /4 ∞ , and ρ , = 11 (ω − E0 )2 + R2 /2 (ω − E0 )2 + R2 /2

(14.62)

and they assume a constant value of 1/2 for R  ω − E0 . The steady-state up and down transition rates between the two levels near the resonance value increases with incident intensity, but the relative increase off-resonance is larger than on resonance since there is more “headroom” for off-resonance growth. A nice early experimental study of power broadening in the optical regime can be found in [21].

14.6 Link of KHD and Low Intensity BR Rates The time dependent BR and perturbative time-independent KHD theories can both describe the transition rates in a two-level atom in the low intensity limit where a closed form solution of the Bloch equations exists as discussed in Sect. 14.5.1. The question arises whether the two theories predict the same behavior and if, yes, how the formalisms can be related. We would expect that the two theories may be equivalent under certain assumptions since at low incident intensities the populations of the lower and upper states do not change significantly. We recall that the KHD perturbation approach treats the populations as binary with ρ11 = 1 and ρ22 = 0 for the excitation step and the reverse, ρ22 = 1 and ρ22 = 1, for the emission step. The binary populations in the KHD theory must therefore not deviate significantly from those predicted by the low intensity BR theory. This is indeed confirmed by Fig. 14.3, where in the resonant excitation step the upper state population ρ22 (t) is seen to remain quite small, ρ22 < 10−2 , for our model parameters and for an incident intensity of I0  100 mJ/cm2 /fs. Considering that the intensity at state-of-the-art synchrotron sources is only of order I0  10−6 mJ/cm2 /fs

720

14 Resonant Non-linear X-Ray Processes in Atoms

as discussed in Sect. 4.2.8, the intensity range covered by the KHD theory is seen to be quite remarkable. We can therefore directly compare the predictions of the BR and KHD theories up to about I0  100 mJ/cm2 /fs. This is facilitated by the existence of an analytical solution of the Bloch equations given in Sect. 14.5.1 which can then be compared to the prediction of the KHD REXS theory derived in Sect. 13.4. Before we formally compare the two theories, let us take a look at their explicit dependence on the incident beam and atom-specific properties. In both formulations we can describe the incident intensity as previously discussed in Sect. 10.2.1 in terms of the number of photons n pk of bandwidth pk in the mode volume V pk = λ3 ω/ pk so that we can rewrite (14.14) in the intensity form I0 =

n pk ω c n pk pk c = . V pk λ3

(14.63)

The response of the two-level atom in the KHD theory can be factored into a photon part described by (14.63) and an atomic part expressed in terms of the dipolar transition width x between the two states given by (14.1) and the atom-specific core hole clock width  listed in Table 10.2. x furthermore links the fundamental photon energy and wavelength to the dipolar atomic transition matrix elements through the fine structure constant, as elegantly expressed by (10.40) and (12.18). In the BR formulation, the key parameter is the Rabi energy which according to (14.38) is a function of both the beam properties, expressed by (14.63), and the atomic transition width x given by (14.1). The key distinction of the two theories is the coherence time t during which the incident beam interacts with the atom. It does not explicitly appear in the KHD theory, and in the following we shall explore its hidden ramification.

14.6.1 The KHD Transition Rates We start by reviewing the expressions for the KHD cross sections and associated transition rates for the case of a two-level atom. The spontaneous XAS cross section ♦ is that previously given by (14.18) or σXAS ♦ = σXAS

λ2  x  2 /4 , 2 π  E +  2 /4

(14.64)

where x is the polarization averaged transition width defined in (14.1) as x =

8π 2 αf ω |b|r ·  p |a|2 . λ2

(14.65)

The total spontaneous plus stimulated REXS cross section is obtained from (13.28) as

14.6 Link of KHD and Low Intensity BR Rates

721

 σREXS 1 x  = 1 + npk σXAS , d 4π 

(14.66)

where we have omitted the Dirac δ-function linking the incident and emitted photon energies in (13.12) which assures energy conservation and in practice does not limit the REXS linewidth to the XAS value  as discussed in Sect. 13.3. The associated XAS and REXS transition rates W are just the cross sections times the incident photon flux. We obtain the following KHD transition rates W (dimension [1/time]), driven by n pk incident photons in the mode volume V pk = λ3 ω/ pk , where pk is the incident bandwidth WXAS =

n pk c x pk  2 /4 σXAS = n pk 2 V pk  2π  (ω − E0 )2 +  2 /4

(14.67)

and KHD: WREXS =

 x  n pk c σREXS = 1+npk WXAS . V pk 

(14.68)

The REXS rate in the KHD theory is time independent and instead depends on the bandwidth pk of the incident radiation. It is typically associated with a box-like mode volume according to pk = λ3 ω/V pk , reflecting a flat-top energy distributions of width pk .

14.6.2 Link of Bandwidth in KHD and Time in BR Rates In the BR formulation, the REXS rate is time dependent and has spontaneous and stimulated contributions according to (14.35) and (14.36) or spon

stim = BR: WREXS = WREXS + WREXS

x 2 ρ22 (t) + R ρ22 (t),  

(14.69)

where according to (14.44) we have at low intensity

       2 /4 R2 t E t ρ22 (t) = 2 .(14.70) 1 + exp − − 2 cos t exp −  E 2 + 2 /4   2  

g(t)

By use of (14.38) for the Rabi energy, or

722

14 Resonant Non-linear X-Ray Processes in Atoms

R2 = n pk

pk x 2π 2

(14.71)

the total BR REXS rate (14.69) becomes

 pk pk x2  2 /4 g(t) n pk BR: WREXS = 1 + n pk 2 2 2 2π  2π   E +  2 /4

(14.72)

which can be directly compared with the KHD expression (14.68) or   pk x2  2 /4 . KHD: WREXS = 1 + npk n pk 2π 2   E 2 +  2 /4

(14.73)

The two expressions are identical for pk =

2π 2  . g(t)

(14.74)

For short times, the expression g(t) may be expanded up to second order to yield pk =

8π 2 2 .  t2

(14.75)

The link between the mode bandwidth pk in the KHD formation and the coherence time t in the BR theory, given by (14.74) is plotted for resonance excitation, E = 0, as a red curve in Fig. 14.10. The short time limit (14.75), shown in gray, intersects the long time limit pk = 2π 2 , shown as a dashed gray horizontal line at the blue vertical demarkation time t = 2τ , where τ = /  = 1.53 fs is the core hole clock time (see Fig. 12.4). From Fig. 14.10 it is apparent that in the long time or steady state limit, an interesting relation exists between the incident mode and atomic emission bandwidth in the form pk = 2π 2 . In the following we will take a closer look at this relationship involving the peculiar factor 2π 2 .

14.6.3 Mode-Based Versus Atom-Based Coherence Volumes In the photon-atom interaction, the number of incident photons n pk of polarization mode p, direction k, and bandwidth pk are assumed to be contained in a quantization box whose volume is given by (3.106) or Vpk = λ3

ω . pk

(14.76)

Mode bandwidth

pk (eV)

14.6 Link of KHD and Low Intensity BR Rates

104 103

Time-bandwidth in KHD and BR theories 2

(a)

10 2 10 1

723

0.1

8

2

2

=0.43eV =1.53 fs

h

t2

2

0.5 1

5 10 Time t (fs)

50 100

Fig. 14.10 Relationship between the mode bandwidth pk of the incident photons in KHD theory and the instantaneous elapsed time t for a two-level atom in the BR theory. The red curve is given by (14.74) for zero detuning, E = 0, and the vertical blue line at t = 2τ separates the short and long time behavior in the BR theory, with the short time response (14.75) shown as a gray line and the long time (equilibrium) response by the dashed gray horizontal line

The key quantity in the quantum mechanical description of the incident beam is the photon density or energy density which because of the constant speed of light, c, is reflected by the expression for the quantum mechanical intensity (10.9) or I0 =

n pk c ω n 0 c ω . = V V pk

(14.77)

Here n 0 is the number of photons of energy ω in the “volume” V . This normalization volume is just an artificial construct, typically taken as a box, that falls out when physical observables are calculated. The intensity I0 may therefore be expressed in different ways by preserving the density n 0 /V = n pk /V pk . It is often convenient to consider another specific “volume”, namely that associated with an atom. Since each resonant electronic decay of energy ω has a definite linewidth , we may write (14.77) also in terms the number of photons n  of bandwidth  contained in the atom-specific volume V as I0 =

n pk c ω n  c ω = . V pk V

(14.78)

The photon density is conserved according to n pk /V pk = n  /V . We obtain the following result [22].

724

14 Resonant Non-linear X-Ray Processes in Atoms

(a) Mode coherence volume Vpk 2

npk coherent photons

h pk

(b) Atomic coherence volume V 2

n coherent photons

h

2

2

Fig. 14.11 Illustration of different longitudinal x-ray coherence volumes with the same lateral coherence area. a The mode coherence volume Vpk containing npk photons is the product of the lateral minimum coherence area Acoh = λ2 and the longitudinal coherence length coh = λ ω/ pk . b In the atomic clock coherence volume V containing n  photons, the longitudinal coherence length is given by coh = λ ω/(2π 2 ), where  is the atom-specific decay width

We can define an atom-specific coherence volume according to V =

ω λ3 . 2π 2 

(14.79)

It contains the number of photons n  of bandwidth  and is linked to the number of photon in mode pk of bandwidth pk by n  = npk

pk . 2π 2 

(14.80)

This equation describes the density-conserving conversion of single mode photons of wavevector k and bandwidth pk into continuous-mode photons of bandwidth  that may propagate into any k direction in the entire 4π solid angle. The link of the numbers npk in volume Vpk and n  in volume V may be envisioned as illustrated in Fig. 14.11. The associated photon fluxes may simply be pictured as the number of photons of bandwidth ω and longitudinal coherence length  = λ ω/ ω that flow through an area λ2 with the speed of light. We can summarize as follows.

14.6 Link of KHD and Low Intensity BR Rates

725

For the description of x-ray interactions with atoms, we may use different normalization volumes that contain the incident photons, defined either by the illuminated sample volume, V , illustrated in Fig. 3.2, or the volumes illustrated in Fig. 14.11. When at low incident intensity the interaction is treated in the time-dependent BR theory versus the bandwidth-dependent KHD theory, care has to be exercised since the time-bandwidth conversion between the two theories is given by (14.74), rather than by a simple linear time-bandwidth uncertainty product. In summary, we have the following relations, Illuminated sample volume: V = A d ω Mode coherence volume: Vpk = λ3 pk ω Atomic coherence volume: V = λ3 2 . 2π 

(14.81)

The photon intensity is then given by the number of photons of energy ω that traverse these volumes at the speed of light c = λω/(2π ) as I0 =

n pk c ω n 0 c ω n  c ω = . = V Vpk V

(14.82)

We can also rewrite the incident photon flux 0 = I0 /ω in (14.14) as 0 =

n pk c pk n  c 2π 2  = . λ3 ω λ3 ω

(14.83)

14.6.4 Zero-Point Field in the Bloch-Rabi Formalism In conjunction with the KHD theory, we have discussed in Sect. 12.2.2.1 how spontaneous x-ray emission can be associated with the zero-point (ZP) field which can be expressed in terms of the mode volume according to (12.3) or k 2 | = |E ZP

pk ω = . 2 0 V pk 2 0 λ3

(14.84)

The same relationship also readily follows from the BR theory. Again, the zero-point field can be derived by comparing the spontaneous and stimulated scattering rates which will be equal if the incident field has the strength of the ZP field. By use of (14.69), the ZP field is determined from the condition

726

14 Resonant Non-linear X-Ray Processes in Atoms spon

WREXS x  = 2 = 1. stim WREXS R

(14.85)

We can now insert the value of the Rabi energy from (14.88) to express the ZP field and obtain it either in the form (14.84) as the number of virtual photons in the volume V pk or equivalently in terms of the number of virtual photons in the volume V and with (14.81) we obtain ω π 2  2 | = = . (14.86) |E ZP 2 0 V 0 λ 3 The energy density of the zero-point field per photon of energy ω is given by ω/V , where V is either V pk or V , so that ZP energy density: EZP =

pk 2π 2  = . λ3 λ3

(14.87)

In the following we will show that the bandwidth relation pk = 2π 2  holds for all steady-state or long time absorption, emission and scattering rates in the BR and KHD theories up to incident intensities where saturation effects occur.

14.7 Link of BR Rates and KHD Rates in the Steady-State We now consider the case of arbitrary incident intensity in the long time limit, where the two-level atom assumes an equilibrium population in the lower and upper states as discussed in Sect. 14.5.4 and illustrated in Fig. 14.8. For this case, the comparison of the BR and KHD theories is again facilitated by the existence of a closed form ∞ by (14.56). expression for the populations, expressed for ρ22

14.7.1 Steady-State Rate Expressions In the long time limit or steady state, all relevant rate expressions may be summarized as follows. The Rabi rate per atom is given by (14.38) and with the new expressions (14.77) and (14.78) we have R2 = I0

pk λ3 x = n   x x = n pk 2π 2 c 2π 2

(14.88)

14.7 Link of BR Rates and KHD Rates in the Steady-State

727

where the dipolar transition width is given by (14.1) and linked to the peak spontaneous XAS cross section by (14.18). We have x =

8π 2 αf ω π ♦ |b|r ·  p |a|2 = 2 σXAS λ2 λ

(14.89)

In equilibrium (long time limit), the excited state population is given by (14.56) or ∞ ρ22 =

n  x  2/4 R2 /4 = (14.90)  (ω − E0 )2 +  2 /4 + R2 /2 (ω−E0 )2 +  2/4 1+2n  x

∞ ∞ = 1 − ρ11 . This compares to the natural Lorentzian lineshape and ρ22

L=

 2/4 . (ω − E0 )2 +  2/4

(14.91)

In all cases the quantities i represent transition energy widths (FWHM, dimension [energy]). In the following we denote the corresponding transition rates as Wi = i / (dimension [1/time]) and denote their low intensity limit for R   by a ♦ superscript. The various rates are given by

Absorption: WXAS =

R2 ∞ ρ ,   11

R 

=⇒

♦ WXAS = n

x 

(14.92)

Spontaneous Auger: WA =

x  A L 

A ∞ ρ ,  22

=⇒

WA♦ = n 

x ∞ ρ ,  22

=⇒

♦ WREXS = n

(14.93)

Spontaneous Scattering: spon

WREXS =

x2 L 

(14.94)

Stimulated Scattering: stim WREXS =

2 R2 ∞ stim ρ22 , =⇒ W ♦ REXS = n 2 x L.  

(14.95)

728

14 Resonant Non-linear X-Ray Processes in Atoms

The REXS expressions correspond to the case where all of the scattered radiation is collected, i.e. when an integration is performed over the emitted photon energy. ∞ ∞ stim = ρ22 = 0.5, we obtain WXAS = WREXS so that the Since at high intensity ρ11 XAS and stimulated REXS rates become the same. At high intensity, R  , all BR rates show a power-broadened behavior on ∞ ∞ or ρ22 as illustrated in detuning energy ω − E0 through their dependence on ρ11 Fig. 14.9. The detuning response is determined by the incident bandwidth which is implicitly assumed to be negligibly small. For the absorption case, the detuning response also determines the measured XAS resonance width. For the REXS case, the linewidth of the emitted photons is not determined by the detuning response but by conservation of the energy bandwidth between incident and emitted photons. The expressions (14.94) and (14.95) correspond to the case where the energy of the emitted radiation is not resolved but integrated over. This corresponds to integration of the Dirac δ-function in (13.12) over ω2 for negligible instrumental resolution and similarly integration over the Gaussian in (13.13) for finite instrumental resolution. In both cases, the integration yields 1. The BR rates reduce to the KHD rates at low intensity with the exception of the XAS lineshape. As shown by (14.92) the low intensity BR rate does not contain the Lorentzian lineshape factor present in the KHD expression (14.67), but the resonant values are the same. With increasing intensity, the BR XAS rate assumes a resonant ∞ (ω) begins to deviate from a constant unit value as shown in lineshape because ρ11 Fig. 14.9a.

14.7.2 Illustration of the BR Rates and Their Saturation As an example, we have plotted in Fig. 14.12 the change of the various rates for the L3 resonance excitation (ω = E0 ) of Co with incident intensity I0 . The curves in Fig. 14.12 were calculated by assuming resonance ω = E0 according to (14.92)– model = 6.25 Mb (see (14.95). We used the parameters √ for the Co L3 resonance σXAS −3 I0 (see (14.39)) and  = 0.43 eV. The steadyFig. 14.2) or R [eV] = 3.8×10 state populations as a function of intensity are those plotted previously in Fig. 14.8. The plotted transition linewidths Wi in units of [eV] correspond to rates Wi where  = 0.658 eV fs. The linewidths calculated by means of the BR theory are plotted as solid lines, those in the low intensity approximation as dashed lines, which also represent the KHD theory, whose validity is emphasized by gray shading. From our results shown in Figs. 14.8b and 14.12 we can state as follows. For a two-level atom, the KHD theory remains valid as long as the Rabi energy R , which according to (14.39) scales with the incident intensity as

14.7 Link of BR Rates and KHD Rates in the Steady-State

729

√

I0 , remains smaller than the natural decay width  of the excited state by R ≤ 0.1. While the spontaneous rates are proportional to the incident intensity, the stimulated REXS rate is proportional to the square of the incident intensity. It becomes equal to the spontaneous rate around 10 mJ/cm2 /fs, meaning that the stimulating incident field is now as strong as the zero-point field, which corresponds to the condition n  = 1, as indicated on top of Fig. 14.12. At an incident intensity of > 1000 mJ/cm2 /fs, the BR widths Wi (solid lines) begin to deviate from the KHD widths Wi♦ (dashed). The stimulated REXS rate becomes equal to the Auger decay rate around 10 J/cm2 /fs, and at this point we have R = . According to (14.56) and Fig. 14.8a, the upper state population has increased to 1/3 and the lower state population is reduced to 2/3. Both spontaneous rates now begin to saturate and even the stimulated rate increases at a reduced rate,

1

10 2

10 4

n

(eV)

ZP field

100

Transition rate widths

1 10

2

10

4

10

6

10

8

10

10

0.1

Co L3-edge XAS spon REXS stim REXS Auger

stim.= spon. REXS

stim. scat. = Auger

1000 10 105 2 Incident intensity I0 (mJ/cm /fs)

Fig. 14.12 Calculated BR widths Wi (solid lines) and low intensity approximation widths Wi♦ (dashed lines) for the equilibrium (long time) case versus incident intensity I0 . The widths were calculated according to (14.92)–(14.95). We have assumed resonance, ω = E0 , and parameters for √ model = 6.25 Mb,  = 3.8 × 10−3 I [eV] and  = 0.43 eV. Note that the L3 -edge in Co metal, σXAS R 0 in the low intensity case the XAS and Auger rates are the same. The stimulated and spontaneous REXS rates become the same at the left vertical line, corresponding to one stimulating photon (n  = 1) in the atomic clock coherence volume. At the second vertical line, the stimulated REXS rate becomes as large as the Auger rate. At larger intensities, the BR XAS (solid black) and stimulated REXS (solid red) rates become the same. The gray shaded region identifies the validity range of the KHD theory

730

14 Resonant Non-linear X-Ray Processes in Atoms

102

Rel. transition probabilities

1

1

spon. XAS

XAS

10

1

10

2

10

3

10

4

10

5

0.1

10 4 n

stim. REXS

Auger spon. REXS

x

10 1000 105 2 Incident intensity I0 (mJ/cm /fs)

Fig. 14.13 Resonant, ω = E0 , relative BR transition probabilities as a function of incident intensity. We have used the same parameters as for Fig. 14.12. The shown transition probabilities are ♦ . Colors distinguish the relative the respective BR rates, normalized to the absorption rate WXAS XAS (black), Auger (gray), spontaneous REXS (blue), and stimulated REXS (red) probabilities. At the bottom we also indicate that it takes n  = / x  103 photons (see Fig. 14.12) to make the stimulated REXS equal to the Auger probability. At the highest intensities the XAS and stimulated REXS probabilities saturate at half the spontaneous XAS value

owing to the fact that the excited state population approaches that of the ground state. At higher intensity, the Auger rate saturates and the absorption rate becomes the same as the stimulated rate. The fact that for the resonant case shown in Fig. 14.12 the solid BR lines are identical to the dashed KHD lines up to a remarkably high incident intensity of 1000 mJ/cm2 /fs, allows us to make the following important statement. For a two-level atom excited at resonance, ω = E0 , the optical Bloch equations and the KHD perturbation theory give the same transition rates over a remarkably large intensity range. The KHD theory only fails at high intensity when the populations in the lower and upper states begin to deviate significantly from the binary values 0 and 1 assumed in the KHD theory. Then the peak value of the populations and their distribution widths change due to power broadening as illustrated in Figs. 14.8, 14.9 and 14.12. Figure 14.12 shows that the spontaneous rate increase does not change at the point ∼ 10 mJ/cm2 /fs where the stimulated rate becomes the same. Only when the stimulated rate becomes as strong as the Auger rate do, we see a significant change in all rates. This is more clearly shown in Fig. 14.13 where all rates are normalized to ♦ . At the highest incident intensities the two dominant rates, the absorption rate WXAS the absorption and stimulated rates, contain half the power each, owing to the fact that the ground and excited states are equally populated.

14.7 Link of BR Rates and KHD Rates in the Steady-State

Transition rate widths

(eV)

10 3

731

stim. REXS. = Auger

1

10

3

10

6

10

9

Action time t 100 fs 1 fs

stim.= spon. scattering

10

12

10

0.1

1000

10 5

7

10

2

Incident intensity I0 (mJ/cm /fs) Fig. 14.14 Transition widths Wi versus incident intensity for action times t = 1 fs (thin lines) and 100 fs (thick lines) according to (14.97). Parameters are the same as for Figs. 14.12 and 14.13. The thick t = 100 fs curves are indistinguishable from the BR curves of the same color in Fig. 14.12 which correspond to t → ∞

14.7.3 Time Dependence of Rates at Resonance For resonant excitation, we can follow the temporal evolution of all BR rates as a function of the “action time” t by use of (14.48) or 2 ρ22 (t) = 2 R 2  + 2R

      1 − cosh(ωR t) + 3 sinh(ωR t) e−3t/(4) ,   4ωR

(14.96)

where ωR is given by (14.47) and ρ11 (t) = 1 − ρ22 (t). The rates in (14.92)–(14.95) then assume the forms R2 A ρ11 (t), WA = ρ22 (t)   x 2 stim = = R ρ22 (t). ρ22 (t), WREXS  

WXAS = spon

WREXS

(14.97)

In Fig. 14.14 we have plotted the XAS, Auger, and spontaneous and stimulated scattering rates as a function of incident intensity for two action times 1 fs and 100 fs. As the pulse length increases one needs higher intensity to achieve the same transition rate. The crossing of the rates with incident power is independent of time.

732

14 Resonant Non-linear X-Ray Processes in Atoms

14.8 Optical Theorem: Sum Rule for Absorption and Scattering In the BR theory the total absorption and scattering rates derived in Sect. 14.7 satisfy an important sum rule that links the two rates. This sum rule is a manifestation of the optical theorem previously discussed in Sect. 7.6 and given by (7.56) or Im f (Q = 0) =

1 σabs k σtot = . (σabs + σscat )  4π 2λ 2λ

(14.98)

Since the essence of the optical theorem is energy conservation, it must hold even in the presence of stimulation. The two-level atom is a particularly simple case since only elastic scattering needs to be considered. The optical theorem can then simply be stated by energy conservation between the different transition probabilities illustrated in Fig. 14.13. The balance between the spontaneous and stimulated contributions of the total XAS and REXS rates is more clearly revealed by the normalized probabilities plotted in Fig. 14.15.

102

1

10 4

Rel. transition probabilities

ZP field: 8 x109 V/m

1.0

n

spon. total XAS

0.1 total REXS

0.01

0.001 0.1

spon.

10

103

2

105

Incident intensity I0 (mJ/cm /fs)

Fig. 14.15 Illustration of the sum rule given by (14.99) for resonant excitation, ω = E0 . Plotted are the relative BR XAS probability (black) and sum of the spontaneous and stimulated BR REXS (red) probabilities, given by the respective BR rates, normalized to the spontaneous (low intensity) XAS rate. We have used the same parameters as for Figs. 14.12 and 14.13 and again identified the validity range of the KHD theory by gray shading

14.8 Optical Theorem: Sum Rule for Absorption and Scattering

733

14.8.1 XAS and REXS Cross-Section Sum Rule The optical theorem (14.98) can be expressed in terms of the relative atomic BR transition probabilities in Fig. 14.15 which account for rebalancing the up and down rates due to population changes. By use of the labels of the black and red curves in the figure we have the probability balance equation spon

spon

spon

− P stim + P + P stim  PXAS . P  XAS  REXS  REXS  REXS

total XAS

(14.99)

total REXS

With increasing incident intensity, the stimulated gain in the REXS probability, stim , is compensated by an equal loss in the XAS probability. Since the areas PREXS under the XAS and stimulated REXS probability distributions are also preserved at all incident intensities, as shown in Fig. 14.9a, we have energy conservation. Note that the Auger rate does not explicitly enter into the sum rule because at low incident intensity it is identical to the (spontaneous) absorption rate, and at high intensity it is negligible. The sum rule holds over the entire intensity range in the BR formalism, which properly accounts for the population changes that lead to a trade-off between the absorption and stimulated channels. Since the sum rule of probabilities represents energy conservation, it can also be phrased in terms of cross sections and similar to (14.98) we have spon

spon

spon

stim +σ + σ stim  σXAS . Atomic sum rule: σXAS − σREXS  

 REXS  REXS

tot σXAS

spon

(14.100)

tot σREXS

spon

The sum rule holds since σXAS  σREXS . The spontaneous XAS cross section is obtained from (14.92) with (14.83) as spon

σXAS =

♦ WXAS λ2  x = . 0 π

(14.101)

The spontaneous parts of the total REXS cross section are obtained from (14.94) with (14.83) as ∞ WREXS λ2 x ρ22 λ2  x  x =  . 0 π  n π  spon

spon

σREXS =

(14.102)

On the far right we have neglected the decrease at high intensity and assumed the ∞ /n  = x /  (see right side of (14.94)). It corresponds to low intensity value of ρ22 the dashed blue line in Fig. 14.15. The stimulated REXS part is derived from (14.95) with (14.83) as

734

14 Resonant Non-linear X-Ray Processes in Atoms stim σREXS =

stim WREXS λ2  x ∞ = ρ . 0 π  22

(14.103)

14.8.2 Atom Transmission Sum Rule The sum rule (14.99) links the total XAS and REXS transition probabilities without considering the directions of the absorbed and scattered photons. We can rewrite the sum rule in terms of the change due to loss and gain in the direction defined by the incident photons. In other words we want to derive a sum rule that reflects photon transmission by an atom in the presence of all linear and NL processes in the BR theory. Such a formulation will help us in the next chapter when we consider the transmission through a solid. There we will see that loss, conventionally attributed to “absorption”, can indeed by compensated by stimulated forward scattering. To see how transparency of an atom arises, we consider a hypothetical incident beam whose lateral size is that of the atomic area, which we will denote as A. We then consider what fraction of the photons going into the atomic area will come back out. We know from Sect. 13.4.1 that the spontaneous REXS signal emitted by the atom consist of two parts containing equal energy. Half the energy is emitted into 4π , the other into the forward direction. Both parts are, however, negligible relative to the spontaneous XAS loss at low incident intensity and stimulated REXS gain at high intensity. Hence we simply ignore it. The transmission loss through the atom will then arise from photons that hit the atom in the fractional atomic area given by the atomic cross section spon

σXAS =

λ2 π 

x  

.

(14.104)

atom loss area probability

As indicated by underbrackets, in this formulation, one may consider the Breit– Wigner cross-sectional area λ2 /π (see Sect. 6.5.1) to represent the atomic area A, and the factor x /  ≤ 1 is just the absorption probability of the incident photons within this area.5 The transmission probability sum rule of a single atom can then be written by denoting loss by a minus sign and gain by a plus sign as atom = 1− Ptrans

π λ2  x π λ2  x  x π λ2  x ∞ + 2 +2 2 ρ 2 π  λ π    λ π  22

λ  spont. loss probability

= 1−

5

spon. gain probability

x x ∞ 2 . + x2 + 2 ρ22   

In the optical region one has x =  as discussed in Sect. 6.5.1.

stim. gain probability

(14.105)

14.8 Optical Theorem: Sum Rule for Absorption and Scattering

735

The factor of 2 in front of the stimulated gain term arises from calling the NL reduction in total “absorption” in (14.100) a stimulated REXS gain in forward scattering. At ∞ ∞ → 0 and at high incident intensity ρ22 → 1/2. low incident intensity we have ρ22

14.8.3 BR Versus KHD Stimulated Enhancement: Saturation The establishment of an equilibrium between the excitation and stimulated decay rates shown in Fig. 14.15, known as saturation of the XAS channel, causes a difference between the KHD and BR theory. This is best seen by expressing the stimulated BR rate in the high intensity limit n  → ∞ in terms of the spontaneous rate. By use of (14.90), (14.94) and (14.95) we find the saturation resonance value  2x 

♦ stim = WREXS WREXS

.

(14.106)

enhancement

The maximum enhancement factor /(2x ) is plotted in Fig. 14.16 for the K-edge of low-Z atoms and the L3 -edge of the 3d transition metals. Since according to Fig. 14.15 the stimulated rate reaches half of the absorption rate at high intensity, one often finds the following expression for the dependence of the stimulated scattering rate on the spontaneous absorption rate and the incident and saturation intensities

Stimulated enhancement in number of photons 3

BR enhancement factor /(2 x )

10

C N O F Ne Na Mg Al Si P S Cl Ar

10 4

2

10 3

10

10 2

10

1

6

8

10

12

14

16

Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn

10 20 18 Atomic number Z

22

24

26

28

30

Fig. 14.16 Maximum enhancement factor at resonance of the stimulated REXS rate over the spontaneous rate in the BR theory according to (14.106)

736

14 Resonant Non-linear X-Ray Processes in Atoms ♦ stim Saturation: WREXS = WXAS

I0 /Isat W♦ −→ XAS , 1 + I0 /Isat 2

(14.107)

where at high intensity n  → /(2x ) or I0 = n 

π 2 c 2 π 2 c 2 −→ Isat = 3 . 3 λ λ x

(14.108)

We note that the enhancement factors shown in Fig. 14.16 do not account for the (much larger) angular enhancement in photon density caused by the confinement of stimulated emission into the forward direction.

14.9 BR, KHD, and Einstein Treatment of a Two-Level System In this section we will compare the BR and KHD treatments of a two-level atom to the famous phenomenological Einstein model put forward 1916/17 to describe the interaction of optical EM radiation with matter [5, 6]. As pointed out already in the Introduction chapter of our book (see Sect. 1.2.2), Einstein’s work, clearly distinguished spontaneous and stimulated emission for the first time and provided a quantum derivation of Planck’s empirical radiation law [23, 24] published 17 years earlier. Below, we start with a short review of the model and then show how it is a special case of the KHD and BR theories that were developed later in time. In fact, one of the early successes of Dirac’s theory [25] was the derivation of Einstein’s spontaneous emission rate.

14.9.1 Einstein’s Model Einstein’s treatment of the interaction of radiation with electronic states is simply amazing because of its simplicity. It completely avoids any specification of the nature of electronic states involved, and through thermodynamic arguments alone succeeds in linking absorption, spontaneous, and stimulated emission rates of a so-called black body, a term coined by Kirchhoff in 1860 [26]. In practice, the blackbody spectrum was defined by the temperature dependent emission spectrum emerging from a small hole in the wall of a platinum (high melting point of 2041 K) box whose interior was originally blackened with iron oxide [27]. The practical implementation of a true blackbody is actually very difficult as reviewed in [28]. The blackbody spectrum covers an energy range that depends on the temperature T of the interior wall of the box. Einstein assumed that the emitted radiation resulted from molecules inside the box which were in thermal equilibrium with the box. The molecules were assumed

14.9 BR, KHD, and Einstein Treatment of a Two-Level System

737

Definition of Einstein rates excited state 22

absorption spontaneous

stimulated

0

B12 *in

A21

ground state

B21 *in

11

Fig. 14.17 Einstein model, balancing “up” and “down” transition rates for a two-level system with dimensionless fractional populations ρ11 and ρ22 , where ρ11 + ρ22 = 1. The rates are given by the Einstein coefficients A21 , and B12 , B21 , with the dimension [1/time], and the incident normalized ∗ = E /E , where E is the energy of the incident EM wave, and E energy Ein in ZP in ZP is the zero-point energy which causes radiative spontaneous decays

to be two-level systems with a continuum of energy separations E0 and populations determined thermodynamically by a Boltzmann distribution. Einstein phenomenologically distinguished three different rates, absorption, spontaneous emission, and stimulated emission, in the two-level molecules that were determined by constant transition matrix elements but depended on temperature through the Boltzmann populations of the levels. The work, which introduced the stimulated emission concept, preceded the invention of the maser and laser by more than 40 years. One may ask why Einstein did not already think of utilizing stimulated emission? In Einstein’s model the maximum intensity or saturation intensity of a blackbody light source is approached when all radiation modes are filled with one photon, each. In practice, blackbody light sources have intensities that are about a factor of 10 less than the saturation intensity, and thus spontaneous emission dominates over stimulated emission. The concept of a laser requires the recognition that one needs to abandon the equilibrium concept and temporarily create a population inversion and extend the two-level concept to more levels with suitable relative decay times. In the following we will briefly review the essence of Einstein’s model by utilizing Fig. 14.17. The difference between the optical and x-ray cases is the prominent role of Auger decays which in the soft x-ray region completely dominate. However, we can simply account for Auger decays by adding its “ghost” linewidth contribution to the radiative spontaneous linewidth as discussed in Sect. 12.2.5. As illustrated in Fig. 14.17, the Einstein model considers three types of first order transition rates in a two-level system. The response to an incident field is expressed as a time-dependent change in the populations of the lower and upper states, which are defined as unitless fractional populations that at any time obey the conservation law

738

14 Resonant Non-linear X-Ray Processes in Atoms

ρ11 (t) + ρ22 (t) = 1.

(14.109)

The population change is driven by excitations from the lower state, i.e. x-ray absorption, and spontaneous and stimulated decays from the upper state. The Einstein model is based on two fundamental assumptions. First, the energy density of the light acting on the two-level system is assumed to be constant over the space of the system and constant in time. The latter means that the system can be assumed to be in equilibrium. Second, the intensity distribution or spectrum of the light is assumed to be smooth and broader than the energy separation E0 between the two levels. The twolevel molecules are therefore immersed in a radiation field, and the Einstein model is designed to find the equilibrium populations in the two-level molecules. The quantities A21 , B12 , and B21 in Fig. 14.17 are called Einstein coefficients. As reviewed by Hilborn [29, 30], they can be found with different dimensions in the literature. Following our formulation of the BR and KHD theories in terms of transition rates, we will also express the coefficients in terms of rates with dimension [1/time]. This is accomplished by defining the energy density of the photon bath by a normalized dimensionless quantity denoted Ein∗ in Fig. 14.17 according to Ein∗ =

Ein |E in |2 = . EZP |E ZP |2

(14.110)

It expresses the incident energy density Ein in units of the zero-point energy density EZP given by (14.87). The equilibrium transition rates are then determined by a rate equation for the population balance given by dρ22 dρ11 = −B12 Ein∗ ρ11 + B21 Ein∗ ρ22 + A21 ρ22 = − . dt dt

(14.111)

The two-level system, assumed to consist of many molecules with different twolevel separations, is immersed in a radiation bath where thermal equilibrium is accounted for by applying a Boltzmann distribution to the level populations. One then finds that the three Einstein coefficients are interrelated and the time-dependent solution of (14.111) can be obtained with A = A21 and B = B12 = B21 . With the initial state condition ρ11 = 1 one obtains the solution ρ22 (t) = 1 − ρ11 (t) =

     BEin∗ 1 − exp − A + 2BEin∗ t . ∗ A + 2BEin

(14.112)

This reduces to the steady-state solution ∞ ∞ = 1 − ρ11 = ρ22

BEin∗ . A + 2BEin∗

For Ein∗ = 0 the excited state population evolves according to

(14.113)

14.9 BR, KHD, and Einstein Treatment of a Two-Level System

which yield the result

739

dρ22 = −A ρ22 dt

(14.114)

ρ22 (t) = ρ22 (0) e−At .

(14.115)

If the excited state decays exponentially with a natural line width  =  A + x , we obtain the expected result, ρ22 (t) = ρ22 (0) e−t/ .

(14.116)

14.9.2 Reduction of the BR to the Einstein Theory Since the Bloch equations cover various scenarios of incident light, ranging from narrow to broad bandwidth, one would expect that the Einstein theory should be recovered in the limit of low incident intensity and broad bandwidth. This is indeed the case. At low intensity where 2R2   2 we can directly compare the analytical results of the BR theory given by (14.44) to the results of the broad bandwidth Einstein theory. If we integrate (14.44) given by ρ22 (t) =

 

 R2 /4 ω−E0 −t/ −t/(2) −2 cos t e 1+e (14.117) (ω−E0 )2 +(/2)2 

over the incident photon energy, corresponding to broadband incident radiation, we obtain ∞ ρ22 (t) dω = −∞

 π R2  1 − e−t/ . 2

(14.118)

We now need to take into account that our energy integration was over a Lorentzian lineshape. To obtain an equivalent expression to the broad flat-top energy distribution assumed in the Einstein theory, our integration result (14.118) needs to be multiplied by a factor 2/(π) (see (10.36) and (10.37)), so that we obtain Low intensity BR : ρ22 (t) =

 R2  1 − e−t/ . 2 

(14.119)

When we compare this result with the Einstein expression (14.112), which for low incident energy density or 2BEin∗  A reduces to

740

14 Resonant Non-linear X-Ray Processes in Atoms

Low intensity Einstein : ρ22 (t) =

 BEin∗  1 − e−A t A

(14.120)

we find equivalence for A=

2 n  x  , and BEin∗ = R = .   

(14.121)

This establishes the connection of the Einstein and BR theories for light of low intensity and broad bandwidth, encountered for conventional light sources.

14.9.2.1

Illustration of Difference of Einstein and BR Theories

The time-dependent solutions of the Bloch equations, integrated over a broad bandwidth window, do not in general merge into those of the Einstein rate equations. The reason is that the Bloch equations allow mixing of the lower and upper state wavefunctions through the coherences, which are left out of the Einstein rate equations. At low incident intensity the mixing contributions due to the coherences are negligible and the BR results merge into the Einstein results when integrated over a large enough bandwidth window as illustrated above. At arbitrary intensities, the Einstein result is given by (14.112), which by use of (14.121) may be rewritten as 2 ρ22 (t) = 2 R 2  + 2R

   2 + 2R2 t . 1 − exp − 



 

(14.122)

This may be compared to the time dependent solution of the Bloch equation, ρ22 (t), determined numerically and then integrated over bandwidth. For the Einstein and ∞ , the energy integrated BR result to have the same long time (equilibrium) value ρ22 numerical BR result needs to be renormalized by multiplying by a factor of f norm =

2 1  . π  1 + 2 2 /  2

(14.123)

R

This renormalization accounts for integration over the Lorentzian-like BR lineshape similar to our previous renormalization of (14.118). The difference between the time dependence of the upper state populations ρ22 (t) in the two theories is illustrated in Fig. 14.18, where the BR bandwidth integration effect is demonstrated for different incident intensities. At the lowest intensities, I0 = 0.1 mJ/cm2 /fs and I0 = 100 mJ/cm2 /fs, the BR theory is seen to seamlessly merge into the Einstein result with increasing bandwidth integration. This is no longer the case for I0 = 105 mJ/cm2 /fs where the strong mixing

14.9 BR, KHD, and Einstein Treatment of a Two-Level System

741

of the lower and upper states by the coherences in the BR theory always causes a deviation from the Einstein theory due to remaining Rabi oscillations. In all cases, the threshold delay of the increase of ρ22 (t) in the BR theory which is present for resonant excitation (gray line) relative to the Einstein theory reduces with increasing bandwidth integration and the equilibrium values at large t become the same.

(a)

6 =0.43eV =1.53 fs

4 I0=0.1 2 mJ/cm /fs

2 0 8 (b) 6

Population

22

(x 10-3)

Population

22

(x 10-6)

Einstein versus BR theory 8

4 I0=100 2 mJ/cm /fs

2

BW= 0 BW= + -1eV BW= + - 5eV BW= + - 20eV

Einstein

0

Population

22

(c) 0.6 0.4 0.2

5

I0=10 mJ/cm2/fs

0.0 0

2

4

6 8 10 Time (fs)

12

14

Fig. 14.18 Comparison of the time dependent populations ρ22 (t) for three different incident intensities, a I0 = 0.1 mJ/cm2 /fs, b I0 = 100 mJ/cm2 /fs, and c I0 = 105 mJ/cm2 /fs. For each intensity the Einstein result, given by (14.122), shown in red is compared √ to the numerical solutions of the Bloch equations for the same parameters, R = 3.8 × 10−3 I0 [eV] and  = 0.43 eV. The resonant (no bandwidth integration) BR result is shown in gray and the bandwidth (BW) integrated and renormalized (multiplication by f norm in (14.123)) results are shown for integration windows BW= ±1 eV in green, BW= ±5 eV in blue and BW= ±20 eV in black. In (a) and (b) the red and black curves are the same as indicated

742

14 Resonant Non-linear X-Ray Processes in Atoms

14.10 Resonance Fluorescence 14.10.1 Introduction The interplay of absorption, spontaneous scattering, and stimulated scattering near a resonance in a single two-level atom inserted into a beam is impossible to measure in a transmission geometry because of the miniscule change of intensity. We shall see in Chap. 15 that this will become possible for samples containing many atoms where the change in signal upon transmission becomes quite large. The dominance of the incident intensity for the single atom case can however be avoided by measurement of the intensity that is spontaneously scattered out of the beam, referred to as “resonance fluorescence” [7, 9]. The discussion of the resonance fluorescence spectrum has a long history, starting with its discussion in 1931 by Weisskopf [31]. This is the spectrum of the spontaneous decay channel whose peak intensity corresponds to the blue line in Figs. 14.12 and 14.13. Resonance fluorescence may be measured in the geometry illustrated in Fig. 14.19 where the incident beam traverses an atomic beam at right angles, whose atoms have been prepared in a suitable two-level state. The incident photon energy ω is typically tuned to the resonance energy of the atoms, and the emitted fluorescence spectrum is measured with a spectrometer/interferometer and a photon detector as a function of the energy difference of the incoming and scattered photons. Since elastic scattering has a node along the polarization vector of the incident radiation, one often chooses the detector (spectrometer) position perpendicular to the E-vector, as shown. In some experiments, an additional coincidence circuit is used to detect time-dependent photon-photon correlations. For a two-level atom one would expect that the fluorescence spectrum is just a peak at the resonance energy E0 . In the KHD theory for example, this is indeed so because the photon and atom parts of the interaction are coupled only by a weak perturbation, so that in the evaluation of the KHD formula the photon and atomic parts can be separated. In the BR theory, the coupling of the photon and electron system is treated to higher order, which is explicitly expressed by the coherences ∗ . Below we will show that at high intensity, the presence of the coherences, ρ12 = ρ21 which were already shown to introduce Rabi oscillations of the populations, leads to a remarkable splitting effect of the spontaneously scattered spectrum, where the splitting itself is directly given by the Rabi energy R . The existence of such a splitting was first pointed out and theoretically derived by Mollow [32] in a pioneering paper in 1969. The splitting occurs only at very high incident intensity, more specifically, when the Rabi interaction energy R of the photon-atom system becomes comparable to the total decay energy width  of the upper state. Semi-classically, the splitting arises from the coupling of a strong oscillating electric field to the electronic system, referred to as the dynamical or AC Stark effect also known as the Autler–Townes effect after Stanley Autler and Charles Townes who discovered the effect in 1955 [33].

14.10 Resonance Fluorescence

743

E incident photons

at be om am ic

Resonance fluorescence arrangement

fluorescent photons spectrometer/ interferometer beam splitter

detector time delay

detector

coincidence counter

Fig. 14.19 Schematic of experimental geometry for resonance fluorescence measurements. The incident laser beam intercepts an atomic beam of low density at right angles. The atomic beam is prepared in a suitable electronic state beforehand, typically though interaction with another laser beam. The incident photon beam, shown to be linearly polarized, is typically tuned to exact resonance of the two-level atoms. The spontaneously scattered fluorescence spectrum is recorded as a function of the energy difference of the incident and scattered photons by use of a spectrometer or interferometer whose optical axis is perpendicular to the laser beam, its polarization axis, and the atomic beam. Also shown is a coincidence circuit that may be employed to investigate temporal correlations of two fluorescent photons as a function of delay time

Mollow’s theory of resonance fluorescence was followed in the 1970s by theoretical [8, 15, 18, 34] and experimental work [11, 35–38]. In particular, Milonni [15] showed that the resonance fluorescence spectrum of a two-level atom may be described equivalently by the semi-classical Bloch equations and the quantum electrodynamics treatment of spontaneous emission. In a special version of the experiment, Kimble, Dagenais, and Mandel [11, 38] used the additional coincidence circuit shown in Fig. 14.19 to demonstrate the existence of so-called photon antibunching in the time-dependent photon-photon correlation signal in resonance fluorescence that cannot be explained classically. Today, the study of strong light-matter interactions, first explored through single atom resonance fluorescence, has been extended by the use of EM radiation in the microwave [39] and hard x-ray regimes [40] and for the study of the collective properties of coupled matter as reviewed in [41]. The work in the microwave regime is particularly novel in that it is performed by use of so-called artificial atoms, a superconducting macroscopic two-level system on a chip, whose quantum behavior is of great importance for the development of quantum information science. In the following we will discuss key aspects of the resonance fluorescence signal, such as the distinction of its elastic and inelastic components through the degrees

744

14 Resonant Non-linear X-Ray Processes in Atoms

of first and second order coherence and the calculation of the splitting. The theory is non-trivial, and we will only outline the derivation of the key results, referring the interested reader to the detailed derivations in references [7, 9, 18, 34, 42]. The x-ray case for SASE XFEL pulses has been theoretically studied by Cavaletto et al. [43].

14.10.2 Second Quantization of the p · A Interaction Hamiltonian We begin by briefly reviewing how the conventional photoelectric interaction Hamiltonian in (9.16) may be written in the so-called second quantization form. We start with the interaction Hamiltonian (14.6) written in terms of the electric field operators according to (3.61) as (14.124) Hint = −e r · (E+ + E− ). In quantum optics it is common to express this Hamiltonian in second quantization form. To do so, we take the energy of the atomic ground state |a as zero and consider the Hamiltonian for the atom, Hat , in the excited state |b of energy ωba plus the interaction Hamiltonian as H = Hat + Hint = ωba |bb| − e r·(E+ + E− ).

(14.125)

The dipole matrix elements of the form b|r|a = a|r|b∗ connect states of opposite parity. We now introduce operators πˆ † that raise the atom from a → b and πˆ that make it decay from b → a as πˆ † = |ba| and πˆ = |ab|.

(14.126)

They have the properties πˆ † |a = |b, π|b ˆ = |a, πˆ † |b = π|a ˆ = 0, πˆ † πˆ = |bb|, πˆ πˆ † = |aa|. (14.127) We also have πˆ † πˆ † = πˆ πˆ = 0 and the closure relation for the two-level system, πˆ † πˆ + πˆ πˆ † = |aa|+|bb| = 1.

(14.128)

The interaction Hamiltonian in the dipole approximation can then be expressed as [42] Hint = −b|e r|a(E+ + E− ) (πˆ † + πˆ ),

(14.129)

14.10 Resonance Fluorescence

745

where we have taken the matrix element to be real.6 The connection with the BR treatment becomes apparent by realizing that (14.129) can be directly expressed in terms of the resonant Rabi energy, defined by (14.11), according to Hint = | f |e r ·(E+ + E− )|i| (πˆ † + πˆ ) =

R † ˆ (πˆ + π). 2

(14.130)

We are now in a position to consider the coherence properties of the spontaneously emitted fluorescent intensity.

14.10.3 The Degree of First Order Temporal Coherence ∗ As the name implies, the off-diagonal elements or “coherences” ρ21 (t) = ρ12 (t) of the density matrix are indeed related to the temporal degree of first order coherence g (1) (t1 , t2 ) of the scattered radiation, whose quantum formulation we have previously discussed in Sect. 4.5. It turns out that the resonance fluorescence spectrum can be obtained by a Fourier transform of the degree of first order coherence, as will be discussed below in Sect. 14.11. For this reason we shall derive it now. The complex degree of first order temporal coherence is defined by (4.84) as the two-field correlation function at the same spatial point. For quasi-stationary light it may be written in the form (4.87) as

g (1) (t) =

E − (t0 )E + (t0 + t) E − (t0 )E + (t0 )

(14.131)

For the case of resonant excitation one in interested in g (1) (t) of the scattered fields. The field operators in (14.131) are then replaced by those of the combined photonatom system defined in Sect. 14.10.2, and one writes [7] g (1) (t) =

πˆ † (t0 )πˆ (t0 + t) . πˆ † (t0 )πˆ (t0 )

(14.132)

An analytical solution for the degree of first order temporal coherence g (1) (t) defined by (14.132) can only be obtained for the case of exact resonance and was worked out analytically by Carmichael and Walls [34]. The key part is the reduction of the two time correlation function in (14.132) to a single time correlation function by means of the quantum regression theorem that only depends on the time difference 6 The relation (14.129) is typically approximated by considering the frequency dependence of the operators. While the electric fields oscillate with the frequency of the incident light, the transition operators πˆ † and πˆ are associated with the frequency ωba of the optical transition. Multiplying out the right side of (14.129), two terms have a slow frequency dependence, where the frequency difference is just the detuning energy (ω−ωba ), while the other two terms have a high-frequency dependence corresponding to transitions that are far off-resonance. Omission of these terms correˆ sponds to the rotating wave approximation with the result Hint = −b|e r|a (E+ πˆ † + E− π).

746

14 Resonant Non-linear X-Ray Processes in Atoms

t. At resonance, g (1) (t) may be expressed in the form [7] Exact resonance: g (1) (t) =

∞ ∗ ) (ρ21 ∞ a1 (t) + a2 (t). ρ22

(14.133)

Here t is the correlation time of two fluorescence fields at the same spatial point, which in principle could be measured by replacing the intensity or photon-photon coincidence circuit in Fig. 14.19 with a Michelson-type interferometer. In practice, one measures the Fourier transform of g (1) (t) which according to the Wiener-Khintchine theorem is the normalized fluorescence spectrum or power spectral density. We will see in the following Sect. 14.11 that the measured resonance fluorescence spectrum contains coherent and incoherent contributions which directly yield the desired coherence information. Here we continue with the determination of the quantities in (14.133). ∞ ∞ and ρ22 for  =  are obtained from (14.58) and (14.55) The long time limits ρ21 as ∞ = ρ21

− iR  R2 ∞ , and ρ = . 22 2  2 + 2R  2 + 2R2

(14.134)

Using the following definition (note difference to (14.47)),  ωR =

R2 −

2 16

(14.135)

the terms a1 (t) and a2 (t) can be expressed analytically as7

7 Our expressions for a (t) and a (t) correspond to the notation of Scully and Zubairy [9]. They 1 2 are equivalent to Loudon’s expressions [7] for α1 (t) and α2 (t)  −iR  i e−3t/(4) α1 (t) = 2 1+ 2 8ωR  +2R       2 2    2 iωR t  2 −iωR t + iωR − e − iωR − e × 4 − 4 (14.136) 2 4 2 4

and α2 (t) = .

i e−3t/(4) e−t/(2) − 2 8ωR



    + 2iωR eiωR t − − 2iωR e−iωR t 2 2

(14.137)

14.10 Resonance Fluorescence

747



   2 − 2/4 ∞ a1 (t) = ρ21 sin(ωR t) e−3t/(4) 1− cos(ωR t)− R ωR

(14.138)

and a2 (t) =

e−t/(2) e−3t/(4) + [4ωR cos(ωR t)+ sin(ωR t)] . 2 8ωR

(14.139)

Hence, at resonance, we obtain the first order correlation function as follows [7, 9]. At exact resonance, the degree of first order temporal coherence in resonance fluorescence is given by

  2 1 t + exp − g (t) = exp[−iωt] 2 2  2 + 2R2   2 (/2 − 2iωR ) 3t + exp −iωR t − 8iωR (3/2 + 2iωR ) 4   (/2 + 2iωR )2 3t − exp iωR t − , (14.140) 8iωR (3/2 − 2iωR ) 4 (1)

where

 ωR =

R2 −

2 . 16

(14.141)

This expression is valid for both R ≥ /4 and R ≤ /4. We shall see that the first term in curly brackets corresponds to the coherent fraction and the last three terms exhibit chaotic light behavior. We have plotted the time dependence of |g (1 )(t)| in Fig. 14.20 and the intensity dependence which enters through R in Fig. 14.21, using parameters for the L3 resonance of a Co atom.8

14.10.4 g (1) (t) From Numerical Solutions of Bloch Equations The analytical expression for g (1) (t) given by (14.140), valid at resonance, depends on a single time. It can also be calculated numerically from the Bloch equations, but this involves two calculations with different initial conditions, reflecting that in 8

The corresponding second order degree of temporal coherence is shown in Fig. 14.25.

748

14 Resonant Non-linear X-Ray Processes in Atoms

0

2

Norm. time t / t 4 6

8

10

1.0 I0

0.8

mJ cm2 fs 2

10 4 10 6 10

0.6 0.4

g (1) (t)

0.2 0.0 1.00 0.50 0.20 0.10 0.05 0.02 0.01

0

2

4

6 8 10 Time t (fs)

12

14

Fig. 14.20 Time dependence of the first order degree of temporal coherence |g (1 )(t)| of the radiation scattered by a resonantly excited Co atom according to (14.140), for three incident intensities. Top on linear scale, bottom on logarithmic scale. We have assumed a two-level Co atom with upper state decay energy width  = 0.43 eV (decay time t = /  = 1.53 fs), a resonance energy of E0 =778 eV (λ = 1.59 nm), and a resonant transition matrix element x = 0.34 meV. The field is assumed to be instantaneously turned on at t = 0

g (1) (t)

1 10

1

10

2

10

3

10

t (fs) 0 1 10 100

4

10

2

3

4 5 6 7 10 10 10 10 10 2 Incident intensity I0 (mJ/cm /fs)

10

8

Fig. 14.21 Intensity dependence of the first order degree of temporal coherence |g (1 )(t)| of the radiation scattered by a resonantly excited Co atom according to (14.140). The field is assumed to be instantaneously turned on at t = 0. We have assumed a two-level Co atom using the same parameters as for Fig. 14.20

14.11 The Resonant Fluorescence Spectrum

749

essence g (1) (t) represents a two time correlation function. It can be shown that we can write (14.133) entirely in terms of two solutions of the numerical Bloch equations for two different initial conditions as follows 

 ∞ ∗ ) 1 2(ρ21 ρ22 (t) + −1 ρ (t) − ρ21 (t) . g (1) (t) = 1 + 21 ∞ ∞ 

 2 ρ22 ρ22  

ρ21 (0) = 1,ρ22 (0) = i/2R ρ21 (0) = ρ22 (0) = 0

(14.142)

14.11 The Resonant Fluorescence Spectrum 14.11.1 Form of the Spectrum Before we formally derive the fluorescence spectrum recorded as a function of energy difference between the resonantly tuned incident photons, ω = E0 , and the scattered photons of energy ωs , we schematically illustrate in Fig. 14.22a the modification of the two-level scheme by the AC Stark effect and the resulting fluorescence spectrum. The key to the modification of the fluorescence spectrum with incident intensity is the dynamic Stark splitting of the upper and lower states, as schematically shown in the figure. The shown level splitting may also be expressed through dressed states in the Jaynes-Cummings model [14, 16]. The resonance fluorescence spectrum is recorded with the incident photon energy tuned to the resonance energy ω0 = E0 as a function of the difference between the scattered energy ωs and incident energy ω0 . At low incident intensity, the spectrum consists of a narrow δ-function-like peak reflecting strict energy conservation in accordance with the REXS expression (13.12). When the√ incident intensity I0 becomes so large that the corresponding Rabi energy R ∝ I0 (see (14.38)) becomes comparable to the natural linewidth  of the upper state, the dynamic Stark splitting of the upper and lower states causes the low intensity single line spectrum (red) to evolve into additional broader components shown in blue and green, known as the Mollow triplet, as schematically shown on the right of Fig. 14.22a. The splitting of the Mollow triplet is seen to be a direct measure of the Rabi energy. In Fig. 14.22b, c we show a modern version of a Mollow triplet spectrum, recorded with microwaves scattered on an artificial atom in the form of a macroscopic superconducting loop interrupted by a Josephson junction, that is inductively coupled to a 1D transmission line [39]. The incident wave in the line is scattered by the artificial atom and can be detected in either the forward or backward direction. In the following we will discuss the derivation of the resonant fluorescence spectrum.

750

14 Resonant Non-linear X-Ray Processes in Atoms

The resonant fluorescence spectrum

(a)

valence

h

0

core

(b)

h

Intensity

R

0

-

+

R

R

h

0

s-

h

R

0

(c)

Fig. 14.22 a Schematic of the dynamic Stark effect splitting of a two-level atom in a strong incident field. The red component is the conventional spontaneous REXS spectrum, as a function of the energy difference ω0 − ωs of the resonantly tuned incident light of energy ω0 = E0 and the spontaneously scattered light of energy ωs , which in principle is a Dirac δ-function but shown with finite width. The blue and green components arise from the Stark splitting by R . b A modern example of the resonant fluorescence spectrum of a strongly driven, so-called artificial atom, in a microwave field of Rabi frequency of 57 MHz corresponding to an incident microwave power of -112 dBm or 6.3 × 10−15 W [39]. Experimental data are shown by circles, and the red curve is a theoretical fit. c Splitting of the fluorescence emission spectrum in (b) as a function of the driving power [39]

14.11.2 Fourier Transform of g (1) (t): The Fluorescence Spectrum The temporal dependence of g (1) (t) can be used to obtain the frequency dependence of the emitted radiation, i.e. the fluorescence spectrum, which is usually measured in the geometry illustrated in Fig. 14.19. The normalized spectrum, referred to as the power spectral density, can be obtained by use of the Wiener-Khintchine theorem as the Fourier transform of g (1) (t). The scattered spectrum of energy ωs is symmetrical around the incident photon energy ω0 and given by

14.11 The Resonant Fluorescence Spectrum

1 G(ωs ) = 2π 

∞

751

  g (1) (t) e−iωs t/ + eiωs t/ dt.

(14.143)

0

It is possible to obtain an analytical form of G(ωs ), but one needs to deal with the zero value encountered in (14.141) which leads to a spectral singularity at R = /4. We shall therefore separately give the solutions for the incident intensity ranges defined by R < /4 and R > /4 [7].

14.11.3 The Low Intensity Spectrum: R < /4 In the low intensity range R < /4 we define the real quantity  ωR =

2 − R2 16

(14.144)

and obtain by evaluation of (14.143) with (14.140) [7] G(ωs ) =

2  2 +2R2



δ(ω0 − ωs ) 

elastic spectrum

+

1 (/2)2 π  (ω0 −ωs )2 + (/2)2  

inelastic spectrum,S1

(/2+ 2ωR )2 1 − 16πωR (ω0 −ωs )2 + (3/4− ωR )2  

inelastic spectrum,S2a

+

(/2− 2ωR )2 1 16πωR (ω0 −ωs )2 + (3/4+ ωR )2  

(14.145)

inelastic spectrum,S2b

and the spectral power density is normalized as ∞ G(ωs ) d(ωs ) = 1.

(14.146)

−∞

In (14.145) we have indicated by underbrackets and labels the four components of the spectrum for later discussion. The designations of the underbracketed expressions as “elastic” and “inelastic” are a consequence of the fact that with increasing intensity the inelastic components dominate and it is seen from Figs. 14.20 and 14.21 that the

752

14 Resonant Non-linear X-Ray Processes in Atoms

properties of the fluorescent radiation change from completely coherent (elastic) at low to incoherent (inelastic) at high intensity.

14.11.4 High Intensity Spectrum: R > /4 At high intensity in the range R > /4 we use the definition of the real quantity  ωR =

R2 −

2 , 16

(14.147)

and the normalized spectral density is given by [7] G(ωs ) =

2 δ(ω0 −ωs )  2 +2R2 

 elastic spectrum

+

1 (/2)2 π  (ω0 −ωs )2 +(/2)2  

inelastic central peak,S1

 3ωR (R2 − 2/2) + (5R2 − 2/2)(ω0 −ωs +ωR )   16π ωR (R2 + 2/2) (ω0 −ωs +ωR )2 +(3/4)2  

+

inelastic side peak,S2a

+ 

 16π

3ωR (R2 − 2/2) − (5R2 − 2/2)(ω0 −ωs −ωR )   ωR (R2 + 2/2) (ω0 −ωs −ωR )2 +(3/4)2 

. (14.148)



inelastic side peak,S2b

The spectral power density is again normalized according to (14.146). The first term in the low and high intensity spectra (14.145) and (14.148) contains a Dirac δ-function. It arises from the fact that the spectra are derived from an analytical g (1) (t) function given by (14.140) that is only valid at strict resonance, corresponding to a vanishingly small incident bandwidth. The spectral δ-function is a consequence of strict energy conservation between the incident and scattered energy in REXS, previously encountered in the KHD REXS cross section (13.12).

14.11.5 Calculated Fluorescence Spectra In Fig. 14.23 we show spectra calculated with (14.145) and (14.148) assuming our Co L3 model parameters for different incident intensities, which are specified through the values of R (see (14.39)). We replaced the δ-functions in (14.145) and (14.148)

14.11 The Resonant Fluorescence Spectrum

753

by finite width Lorentzians of FWHM 0 = 0.1 eV with the same unit integrated area according to δ(ω0 −ωs ) ↔

(0 /2)2 2 . π 0 (ω0 −ωs )2 +(0 /2)2

(14.149)

Our chosen Lorentzian lineshape of the elastically scattered component is arbitrary but serves to illustrate the change of the total elastic intensity given by the area under the curve. At low intensity R < /4 the spectrum is dominated by the coherent elastic peak, as illustrated in Fig. 14.23a, calculated with (14.145) at the upper incident intensity √ limit R = /4.01. The curves shown in Fig. 14.23b correspond to R = / 2 where the elastic (coherent) and inelastic (incoherent) integrated spectral intensities are equal. At higher incident intensity the elastic spectrum (red) further decreases as shown in Figs. 14.23c, d, and for R = 3 three inelastic peaks have emerged that are often referred to as the Mollow triplet [7, 32, 42]. The elastic red peak in the panels of Fig. 14.23 contains the following fraction of the total intensity (a) 89%, (b) 50%, (c) 33%, (d) 5% and the intensities in [mJ/cm2 /fs] corresponding to the R values are (a) 814 , (b) 6.5 ×103 , (c) 1.3 ×104 , (d) 1.2 ×105 (also see Fig. 14.24b).

14.11.6 Coherent and Incoherent Parts of the Equilibrium Spectrum The total spectra in Fig. 14.23 represented by the black curves may be separated into coherently and incoherently scattered components. The coherent components correspond to the red curves and the incoherent components to the sum of the blue and green curves. In the long time limit t → ∞ where equilibrium is established one can derive simple expressions for the integrated coherent and incoherent intensities. Since the total integrated spectral densities (black curves) are normalized to unity according to (14.146), the total normalized scattered rate is ∞ Rtot =

G(ωs ) d(ωs ) = 1.

(14.150)

−∞

The coherent contribution is the first term 2 Rcoh = 2  +2R2

∞ δ(ω0 −ωs ) d(ωs ) = −∞



 1



2  2 +2R2

(14.151)

754

14 Resonant Non-linear X-Ray Processes in Atoms

6 5

(a)

R=

3

total elastic inel. S1 inel. S2

4 3

R=

(c)

2

Spectral density

2 1

1

x3

0 x3

1

0

4 (b)

R=

1.0

3

0.8

2

0.6

(d)

R=

R

0.4 1 0.2

x3

0

x3

2

1

0

S2a

S2b

0.0

1

h

2

0

-h

s

2 (eV)

1

0

1

2

Fig. 14.23 Spectral density of the scattered radiation G(ωs ) given by (14.145) and (14.148) for low and high incident intensities, with the Dirac δ-function replaced by a Lorentzian of FWHM 0 = 0.1eV. The black curves represent the total scattered spectra with integrated unit normalization according to (14.146). The different incident intensities correspond to the four R values in units of  = 0.43 eV (Co L3 resonance). We assumed an incident resonantly tuned photon energy of ω0 =778 eV (λ0 = 1.59 nm) and a resonant transition matrix element x = 0.34 meV. The red curves are the elastically scattered components. The blue curve is the central inelastically scattered component S1 and the green curves the sum of the inelastically scattered components S2 = S2a √+ S2b . a Calculated with (14.145) and R = /4.01. b–d Calculated with (14.148) and R = / 2, R = , and R = 3, respectively

and the incoherent contribution is just the difference Rinc = Rtot − Rcoh = 1 −

2 .  2 +2R2

(14.152)

We can now revisit the different rates plotted in Fig. 14.12, which correspond to resonant excitation and equilibrium (t → ∞). The absorption rate shown in black and stimulated rate shown in red correspond to changes of the incident photon rate in the forward “in-beam” direction, and they remain the same as in Fig. 14.12. The resonant fluorescence rate measured in the 90◦ “out-of-beam” direction corresponds to the spontaneously scattered intensity shown as a blue curve in Fig. 14.12. We are now able to quantitatively decompose this curve into coherent and incoherent spectral components shown in Fig. 14.23. The spontaneous scattering rate

14.11 The Resonant Fluorescence Spectrum

10

(a)

755

Co L3-edge

Transition linewidths (eV)

0.1 10

3

10

5

10

7

10

9

abs

stim coh

inc spon

1 (b)

coh

spon

0.1 10

2

10

3

10

4

R

inc

1

=4

2

spon

10 100 1000 104 105 106 2 Incident intensity I0 (mJ/cm /fs)

Fig. 14.24 a Intensity dependence of the various transition rates i / as in Fig. 14.12. The transition energy width i was calculated for resonant excitation (ω = E0 ) and equilibrium (t → ∞). The absorption (black) and stimulated (red) rates are the same as in Fig. 14.12. The spontaneous scattering rate spon (blue) was calculated according to (14.154) and its coherent component (solid purple) from (14.155) and the incoherent component (dashed purple) from (14.156). We have assumed a two-level Co atom with upper state decay energy width  = 0.43 eV (decay time t = /  = 1.53 fs), a resonance energy of E0 =778 eV (λ = 1.59 nm), and a resonant transition matrix element x = 0.34 meV b Relative fractions of the elastic and inelastic scattering rates contributing to the total rate, as indicated. In gray we have indicated by four vertical lines the specific incident intensities in Fig. 14.23 corresponding to R in units of 

spon

WREXS = spon / is given by (14.94) or spon = x

R2  2 + 2 2   R

(14.153)

∞ ρ22

∞ where ρ22 is the equilibrium resonant excited state population and x the dipolar transition width. The total spontaneous rate, spon /, is just the dimensionless unit∞ , i.e. normalized total scattering rate Rtot times a the scaling factor x ρ22 ∞ spon = x ρ22 Rtot .

(14.154)

756

14 Resonant Non-linear X-Ray Processes in Atoms

Similarly, the coherently scattered rate is obtained from (14.151) by the same scaling factor as ∞ Rcoh = x coh = x ρ22

 2  ∞ 2 ρ R2 22

(14.155)

and the incoherently scattered rate is obtained from (14.152) as  2 ∞ ∞ ∞ Rinc = x ρ22 inc (ω) = x ρ22 1 − 2 ρ22 . R

(14.156)

We can now replot Fig. 14.12 by explicitly showing the coherent and incoherent component of the spontaneous rate with the absorption and stimulated scattering rates being the same as in Fig. 14.12. The dependence of the various rates on incident intensity is plotted in Fig. 14.24.

14.12 Second Order Coherence of the Fluorescent Photons The coincidence circuit in Fig. 14.19 may be used to obtain information on photonphoton correlations in the resonance fluorescence signal. In the second quantization formulation (see Sect. 14.10.2), the degree of second order coherence in resonance fluorescence is given by [7] g (2) (t) =

ˆ 0 + t)π(t ˆ 0 ) πˆ † (t0 )πˆ † (t0 + t)π(t . † 2 πˆ (t0 )πˆ (t0 )

(14.157)

It can be shown that for a two-level system g (2) (t) is linked to the excited state population ρ22 (t) by the general relationship [7, 12, 18, 42] g (2) (t) =

ρ22 (t) ∞ . ρ22

(14.158)

While the general solution is obtained by numerical solution of the Bloch equation, it is instructive to consider cases where numerical solutions of the Bloch equations exist as discussed in Sect. 14.5. We shall now illustrate the evolution of g (2) (t) for three cases with analytical solutions.

14.12 Second Order Coherence of the Fluorescent Photons

757

14.12.1 Weak Incident Beam We start with the solution of the Bloch equations for a weak beam given by (14.44). It is derived under the assumption that the incident photons are in the same mode ( p, k), i.e. have the same polarization, propagation direction k, and same photon energy ω =  c k. In practice, the incident photons have a finite bandwidth and we start by assuming that it is very small relative to the natural atomic resonance width given by , so that we can study the influence of detuning of the incident photon energy ω from the exact resonance energy E0 . The second order degree of temporal coherence is then obtained from (14.44) as g (2) (t) = 1 + e−t/ − 2 cos



 ω−E0 t e−t/(2) 

(14.159)

which at resonance, ω−E0 = 0 assumes the simple form 2  g (2) (t) = 1 − e−t/(2) .

(14.160)

We have also seen in Sect. 14.9.2 that (14.159), when integrated over photon energy, is equivalent to the Einstein result given by (14.119). In this case we simply have g (2) (t) = 1 − e−t/ .

(14.161)

In Fig. 14.25a we have plotted the temporal evolution of the degree of second order temporal coherence, g (2) (t) for weak incident intensities fulfilling the condition 4R <  =  (see (14.47)). The red curve is given by (14.159) assuming resonant, E = ω−E0 = 0, while the blue curve corresponds to detuning by E = 2. We also show in black the corresponding Einstein result (14.161), which represents the case of broad bandwidth (short coherence time) incident radiation. We have assumed the parameters given in the caption.

758

14 Resonant Non-linear X-Ray Processes in Atoms

2 nd order degree of temporal coherence g (2) (t)

0

2

2.0 (a)

Norm. time t / t 4 6

8

10

mJ (a) I0 = 10 cm2 fs 2

1.5 1.0 0.5 0.0 2.0

= =

(b)

(b) I0 = 10

1.5

5

mJ cm2 fs

Einstein ~h

1.0 0.5 0.0 0

2

4

6 8 10 Time t (fs)

12

14

Fig. 14.25 a Second order degree of temporal coherence g (2 )(t) for an incident intensity, I0 = 100 mJ/cm2 /fs as a function of relative arrival time (same pathlength) of transmitted and scattered photons by a driven atom. We have assumed a two-level Co atom with upper state decay energy width  = 0.43 eV (decay time t = /  = 1.53 fs), a resonance energy of E0 =778 eV (λ = 1.59 nm) and a resonant transition matrix element x = 0.34 meV. The red curve is calculated with (14.159) assuming resonant excitation, E = ω− E0 = 0 and the blue curve with non-resonant excitation given by E = 2. The corresponding Einstein result shown in black is calculated with (14.161). b Same as in (a) with I0 = 105 mJ/cm2 /fs. The on-resonance curve can be calculated analytically with (14.164), while the off-resonance curve was calculated by numerical solution of the Bloch equations

14.12.2 Large Dephasing: Einstein Result More insight into the evolution of the second order degree of coherence from the narrow bandwidth BR to the broad bandwidth Einstein theory is provided by the weak field solution (14.43) in the presence of a dephasing contribution which adds to the natural linewidth  according to  = +deph . In this case we have with E = ω−E0

14.12 Second Order Coherence of the Fluorescent Photons

759

  2(− /2) E 2 +( /2)2 −t/   e g (t) = 1 +  E 2 + (− /2)2   E t − t/2 2[ (− /2) + E 2 ]/2   cos − e   E 2 + (− /2)2   E t − t/2 2 E (− )  sin +  2 . (14.162) e   E + (− /2)2 (2)

Since the Einstein theory corresponds to broadband excitation, we assume that the dephasing term completely dominates and with   , E we obtain (2)

g (t) = 1 − e

−t/

    2 E t E t 2 E − cos + 2 sin e− t/2   ( /2) 

 1 − e−t/

(14.163)

which merges into the Einstein result (14.161) for large dephasing. This can be readily envisioned as the change of the red and blue curves to the black curve in Fig. 14.25a.

14.12.3 Resonant Case of Arbitrary Intensity While the weak field expression for g (2) (t) is independent of the incident intensity as long as 4R <  =  (see (14.47)), we now explore its changes at higher intensities. In this case an analytical solution to the Bloch equations only exists at exact resonance, but we will also show results obtained by numerical solution of the Bloch equations. We start with the analytical expression for exact resonance excitation given by (14.48), from which we obtain       3 −3t/(4)   sinh(ωR t) e g (t) = 1− cosh(ωR t)+ , 4ω (2)

(14.164)

R

where the absolute value is required since the quantity 1 ωR = 



2 − R2 16

(14.165)

may be imaginary. The results for high incident intensity, I0 = 105 mJ/cm2 /fs, and resonant excitation given by (14.164) are shown in red in Fig. 14.25b. They are compared to the Einstein result (black) and the numerical solution of the Bloch equation for a detuning energy of E = 2 (blue).

760

14 Resonant Non-linear X-Ray Processes in Atoms

In practice, the temporal evolution of g (2) (t) measured with monochromatic light rather than the broadband light assumed in the Einstein model exhibits a rich oscillatory structure with a range of values from 0 to around 2, as shown by the red and blue curves in Fig. 14.25. The Rabi oscillations are due to the fact that the excited and ground state populations oscillate in time with opposite phases (since ρ22 + ρ11 = 1) before they become equal at long times. At delay times t  10 t the second order degree of coherence settles down to unity. This does not mean that the fluorescence light is second order coherent, but rather that it becomes essentially uncorrelated and the two-photon coincidences are accidental. As discussed in Sect. 5.4, true second order coherence requires that both g (1) (t) and g (2) (t) are unity for the same time t. In Fig. 14.26 we illustrate the rich structure of g (2) (t, E) in a 3D plot as a function of time and detuning energy for the same incident intensity of I0 = 105 mJ/cm2 /fs as in Fig. 14.25b.

)

(eV gy

ner

tun

e ing

De

1

2

I0 = 10

5

mJ cm2 fs

0

-1

-2

2 (2)

g (t ) 1

2 0 10

1 0

Tim 5 et (fs)

-1 0 -2

Fig. 14.26 Three-dimensional plot of the second order degree of temporal coherence g (2) (t) of a Co atom for I0 = 105 mJ/cm2 /fs as in Fig. 14.25b. The figure illustrates the rich structure of g (2) (t, E ) as a function of time and detuning energy -2eV ≤ E ≤ 2 eV ( 5), the latter dependence being symmetrical about E = 0

14.12 Second Order Coherence of the Fluorescent Photons

761

14.12.4 Photon Antibunching in Resonance Fluorescence Remarkably, the degree of second order temporal coherence of the elastically scattered radiation in the presence of stimulation evolves from zero, at t = 0 to unity at t  10t . The value of g (2) (t = 0) = 0 has received a lot of attention since it violates the classically allowed range of values 1 ≤ g (2) ≤ ∞ [18]. This result was first experimentally demonstrated by Kimble et al. [11] using a very dilute atomic beam and more recently with single trapped 24 Mg+ ions by Höffges et al. [44]. In particular, the zero value of g (2) (t = 0) is due to the inability of the single two-level atom to emit a pair of photons simultaneously. It was one of the first illustrations of states of light that can only be explained by quantum theory. During the emission of a photon, the atom decays into its ground state and needs to be re-excited before it can emit a second photon. The emitted photons are therefore spaced out in time and are said to be “antibunched” with an average separation of the excited state lifetime / . Figure 14.27 illustrates the opposite signature of “bunching” and “antibunching” in g (2) (t). The red curve corresponds to the photon-photon coincidence count rate as a function of their relative photon arrival for light produced by a chaotic lightsource with a Lorentzian frequency distribution (exponential decay). At short times, photons arrive more likely together, i.e. are “bunched”, than at longer timescales. This is the photon bunching effect observed by Hanbury Brown and Twiss [45–47]. The black curve is taken from Fig. 14.25 (Einstein model) and represents photon-photon coincidence of resonant fluorescence light. It shows the opposite behavior or “antibunching” at short times. The two curves are reflections of each other around the horizontal line g (2) (t) = 1 and the bunching property of chaotic light mirrors the antibunching property of light of an atom driven by a strong field.

2.0 photon bunching

g (2) (t)

1.5

chaotic light with Lorentzian spectrum

1.0 0.5

photon antibunching

resonance fluorescence light (Einstein model)

0.0 0

2 4 6 8 10 12 Correlation time t (arb. units)

14

Fig. 14.27 Second order degree of temporal coherence g (2) (t) as a function of relative photonphoton arrival time for two different types of light. The red curve corresponds to chaotic light with a Lorentzian spectral distribution, while the black curve is taken from Fig. 14.25 and corresponds to resonant fluorescent light emitted by a strongly driven atom

762

14 Resonant Non-linear X-Ray Processes in Atoms

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

41. 42.

I.I. Rabi, Phys. Rev. 51, 652 (1937) F. Bloch, Phys. Rev. 70, 460 (1946) L. Allen, J.H. Eberly, Optical Resonance and Two Level Atoms (Wiley, New York, 1975) R.P. Feynman, F.L. Vernon, R.W. Hellwarth, J. Appl. Phys. 28, 49 (1957) A. Einstein, Verh. Deut. Phys. Ges. 18, 318 (1916) A. Einstein, Phys. Z. 18, 121 (1917) R. Loudon, The Quantum Theory of Light, 3rd edn. (Clarendon Press, Oxford, 2000) H.J. Kimble, L. Mandel, Phys. Rev. A 13, 2123 (1976) M.O. Scully, M.S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997) L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995) H.J. Kimble, M. Dagenais, L. Mandel, Phys. Rev. Lett. 39, 691 (1977) H. Paul, Rev. Mod. Phys. 54, 1061 (1982) J.J. Sakurai, Modern Quantum Mechanics, Revised (Addison-Wesley, Reading, Mass., 1994) C.C. Gerry, P.L. Knight, Introductory Quantum Optics (Cambridge University Press, Cambridge, 2005) P.W. Milonni, Phys. Rep. 25C, 1 (1976) E.T. Jaynes, F.W. Cummings, Proc. IEEE 51, 89 (1963) J. Stöhr, H.C. Siegmann, Magnetism: From Fundamentals to Nanoscale Dynamics (Springer, Heidelberg, 2006) R. Loudon, Rep. Prog. Phys. 43, 913 (1980) N. Rohringer et al., Nature 481, 488 (2012) N. Hartmann, G. Hartmann, R. Heider, M.S. Wagner, M. Ilchen et al., Nat. Photonics 12, 215 (2018) M.L. Citron, H.R. Gray, C.W. Gabel, J.C.R. Stroud, Phys. Rev. A 16, 1507 (1977) J. Stöhr, A. Scherz, Phys. Rev. Lett. 115, 107 402 (2015) M. Planck, Verh. Deut. Phys. Ges. 2, 202 (1900) M. Planck, Verh. Deut. Phys. Ges. 2, 237 (1900) P.A.M. Dirac, Proc. Roy. Soc. A 114, 243 (1927) G. Kirchhoff, Ann. Phys. Chem. 109, 275 (1860) O. Lummer, F. Kurlbaum, Verh. Deut. Phys. Ges. 17, 106 (1898) P.M.L. Robitaille, Prog. Phys. 4, 3 (2009) R.C. Hilborn, Am. J. Phys. 50, 982 (1982) Hilborn, A 2002 update of the original 1982 paper by Hilborn (Am. J. Phys. 50, 982 (1982)) is available by the same author on Arxiv V. Weisskopf, Ann. Phys. Leipzig 9, 23 (1931) B.R. Mollow, Phys. Rev. 188, 1969 (1969) S.H. Autler, C.H. Townes, Phys. Rev. 100, 703 (1955) H. J. Carmichael, D. F. Walls, J. Phys. B: At. Mol. Phys. 9, 1199 (1976); erratum ibid. 9, 2755 (1976) F. Schuda, J.C.B. Stroud, M. Hercher, J. Phys. B 7, L198 (1974) R.E. Grove, F.Y. Wu, S. Ezekiel, Phys. Rev. A 15, 227 (1977) H.J. Kimble, L. Mandel, Phys. Rev. A 15, 689 (1977) H.J. Kimble, M. Dagenais, L. Mandel, Phys. Rev. A 18, 201 (1978) O. Astafiev, A.M. Zagoskin, A. A. Abdumalikov. Jr., Y.A. Pashkin, T. Yamamoto, K. Inomata, Y. Nakamura, J.S. Tsai, Science 327, 840 (2010) R. Röhlsberger, J. Evers, S. Shwartz, Quantum and nonlinear optics with hard x-rays. in Synchrotron Light Sources and Free-Electron Lasers, ed. by E. Jaeschke, S. Khan, J. Schneider, J. Hastings (Springer, Cham., Heidelberg, 2020), p. 1399 D.S. Dovzhenko, S.V. Ryabchuk, Y.P. Rakovich, I.R. Nabiev, Nanoscale 10, 3589 (2018) D. A. Steck, Quantum and Atom Optics. Available online at http://steck.us/teaching (2017)

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43. S.M. Cavaletto, C. Buth, Z. Harman, E.P. Kanter, S.H. Southworth, L. Young, C.H. Keitel, Phys. Rev. A 86(033), 402 (2012) 44. J.T. Höffges, H.W. Baldauf, T. Eichler, S.R. Helmfrid, H. Walther, Opt. Commun. 133, 170 (1997) 45. R.H. Brown, R.Q. Twiss, Nature 178, 1046 (1956) 46. R.H. Brown, R.Q. Twiss, Proc. Roy. Soc. Lond. A 242, 300 (1957) 47. R.H. Brown, R.Q. Twiss, Proc. Roy. Soc. Lond. A 243, 291 (1958)

Chapter 15

Non-linear Absorption and Scattering Processes in Solids

15.1 Introduction and Chapter Overview 15.1.1 Brief Introduction In the optical regime, the advent of the laser required the consideration of non-linear interactions with matter beginning in the 1960s. Over the years they have led to the new field of non-linear optics [1, 2] or strong-field physics [3]. In the x-ray regime, non-linear phenomena naturally became important and have been explored with VUV and x-ray free electron lasers. Of particular interest are spectroscopic x-ray studies of the modification of the electronic structure of solids [4–7], x-ray transparency in non-resonant absorption [8–11] and resonant x-ray absorption [7, 12–14], and the related effects of stimulated x-ray emission [15–20] and stimulated resonant inelastic scattering [21–23]. We also discuss the limits of resonant diffraction imaging induced by sample damage [24] and stimulated forward scattering [25, 26]. There are many other aspects of non-linear x-ray phenomena not covered here as outlined in Sect. 1.5.4.

15.1.2 Chapter Overview The goal of the present chapter is to develop a description of the most fundamental non-linear response of a solid, namely the change in x-ray absorption with incident intensity. Since x-ray absorption has the largest interaction cross section, its understanding is fundamental for all XFEL experiments. X-ray absorption is also the simplest process to quantify since one simply has to measure the change of x-ray intensity upon transmission through a film. In the past we have relied on the fact that even x-ray pulses of modern synchrotron radiation sources typically give information of the “as-prepared” state of a sample of interest. The “as-prepared” state may be any state of a solid that can be stabilized © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_15

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over the effective measurement time. In conventional experiments it is the time the sample is exposed to the beam. In pump-probe experiments it is given by the individual 100 ps pulse length of storage rings. The conventional key paradigm of x-ray science has been that x-rays can be viewed as a weakly interacting probe that provides information of the as-prepared state of the sample before the measurement. Relative to conventional experiments with synchrotron sources, there are two fundamental differences in XFEL pulse interactions with matter. The number of photons per pulse is increased and the pulse length shortened by similar factors of order 104 −105 . At best, XFEL pulses allow us to take ultrafast snapshots on timescales faster than the picosecond atomic motion and comparable to the femtosecond time evolution of chemical bonds. At worst, the intense XFEL pulses will modify the electronic properties even during a single pulse, so that the pulse average already has an electronic damage component, and crystallographic damage will be observed when multiple consecutive pulses are used. It is therefore of fundamental importance to understand when the paradigm of measuring the “as-prepared” state of interest breaks down. In the first part of this chapter we will explore the “damage” problem by discussing the temporal evolution of changes in solids triggered by energy transfer from the incident photons to atoms. This entails understanding how the largest interaction process, x-ray absorption, changes as a function of intensity and the temporal evolution of the energy transfer of the incident photons to the electronic system and the lattice in solids. We are particularly interested in exploring changes in the electronic structure which happens on the shortest time scales. We will show how such changes are reflected by changes of the x-ray absorption spectrum at relatively low incident intensities. The central part of this chapter deals with the description of fundamental nonlinear effects that appear in x-ray absorption at higher incident intensities. We pay particular attention to the description of stimulated resonant forward scattering in solids, building on the previous chapter where the phenomenon was treated for atoms. Our treatment of solids also builds on our previous discussion of their semi-classical response in Chap. 7, and their quantum treatment by the KHD theory, discussed for XAS in Chap. 10, and REXS and RIXS in Chap. 13. In the process we emphasize analytical solutions that allow us to readily treat various cases. Besides treating the change in absorption in the presence of stimulated REXS, we will also consider the effect of increased in-beam forward scattering on diffraction. Since diffraction is based on the out-of-beam scattered intensity, we will show that it is strongly affected by the enhancement of the in-beam forward scattered intensity and may vanish altogether. Besides conventional diffraction by charge domains we also consider the dichroic diffraction of magnetic domains which will be shown to exhibit particularly large non-linear effects. Finally, we will consider the case of stimulated inelastic resonance scattering, stimulated RIXS. We will show that the main advantage of stimulated RIXS comes from the fact that stimulation is directional, so that the scattered photons produced by stimulation are driven into the small solid angle of the detector, which may lead to enhancements of the order of 105 . In practice, however, the intensity required for stim-

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ulated RIXS also causes electronic changes even during ultrashort few-femtosecond pulses. At present, this appears to be a show stopper for stimulated RIXS in solids.

15.2 The Fundamental Damage Issue of XFEL Radiation Historically, the use of x-rays for the study of the electronic and atomic structure of matter was based on the premise that x-rays provide information on the sample in the “as-prepared” state, which is either the electronic ground state at the temperature of interest or a state which has been prepared by application of an external stimulus like pressure, fields, or an electronic excitation by another source, e.g. an optical laser. Until the XFEL era, sample perturbations by the x-ray beam itself could typically be ignored and the interaction treated as a small perturbation, e.g. by the lowest order KHD perturbation theory as outlined in Chap. 9. Early considerations of the effect of XFEL pulses on matter [27] already emphasized potential problems associated with changes of the state of the sample during or after the x-ray pulses. One of the key questions concerns the time scales associated with typical XFEL pulses (see Sect. 2.8) relative to the intrinsic timescales associated with charge, spin, and atomic motions in matter, illustrated in Fig. 1.12. If the pulses are much shorter than the temporal changes in matter, the pulse modifications can be either neglected since they have not materialized during the pulse or, provided that a time zero can be established, the temporal evolution of matter can be studied in detail. For comparable timescales of pulses and processes in matter, photons in the second half of the pulse will already produce a response that differs from the “as-prepared” state. We know typical motion times of atoms, since they determine the speed of sound. More generally, the concept of inertia leads to the motto “the smaller (less mass), the faster”, as illustrated in Fig. 1.12. This concept underlies the Born–Oppenheimer principle, stating that the lighter electrons move faster than the heavy nuclei. For the same reason, it was suggested already before XFELs came into existence [28] that ultrashort XFEL pulses of 1–10 fs length can traverse a sample fast enough that the recorded picture will show the “as-prepared” atomic structure before pulse-induced atomic motions can manifest themselves. In other words, one may beat the speed of sound with the speed of light (more precisely, with the speed of a femtosecond x-ray pulse). On the other hand, mass scaling predicts that this argument does not apply to the electronic structure of a sample. The valence charge distributions of the thousand times lighter electrons, that form the all-important bonding orbitals between atoms, may change on the same timescale as the fastest x-ray pulses. In the following we will consider typical intensities and XFEL pulse lengths to elucidate when “damage” in one form or another will be observed.

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15 Non-linear Absorption and Scattering Processes in Solids

15.2.1 X-Ray Beam Parameters We characterize the x-rays by three parameters, the photon density n ph /V , the frequency ω = 2πc/λ, and the pulse length τ , which we assume to be the same as the coherence time. Together with the constant speed of light c, these quantities define the cross-sectional area of the beam A = V /(cτ ), and with the addition of Planck’s constant , they define the quantum mechanical flux, fluence, and intensity (see Sect. 3.6.2), as n ph c (15.1) Flux :  = V Fluence : F =

n ph ω cτ n ph ω = A V

(15.2)

Intensity : I =

n ph ω n ph ω c = . Aτ V

(15.3)

With our assumption that the pulses are temporally coherent (transform limited), the intensity is simply the fluence divided by the pulse length. We note that this is typically not fulfilled for SASE XFEL sources where the coherence time is determined by the length of coherent subpulses (spikes) within the total pulse (see Figs. 2.29 and 2.30). For coherently driven processes like stimulated REXS it is important to distinguish pulse length and coherence time (see Sect. 15.6.3). In the following we will address the question up to which fluence or intensity one can expect to probe a largely unperturbed electronic state and lattice of a solid sample. Since the answer to this question depends on the type of material and the XAS cross sections of the atoms, we will consider a metal and assume an x-ray absorption cross section of about 5 Mb/atom, which as shown in Fig. 1.15 is characteristic of strong absorption.

15.2.2 Temporal Evolution of Matter after X-Ray Excitation Simple arguments, already confirmed by ultrafast optical laser studies [29, 30], naturally explain the temporal evolution of matter after a severe instantaneous energetic impulse. The photon energy is first transferred to the more agile electronic system of the sample. The energy in the electronic system is then transferred to the atomic phonon system, heating up the lattice. The timescale of electronic and lattice modifications relative to the x-ray pulse length lies at the heart of all XFEL experiments since it directly determines when the conventional “weak perturbation” picture breaks down. The change in matter may manifest itself over a wide range from mild perturbation to complete destruction. An example of a mild change would be an increased electronic or lattice temperature. The other extreme is a transfer of energy of tens of eV to all atoms in the sample, resulting in a plasma like state and complete destruc-

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tion of the sample. This case is itself interesting since the transiently created state is of interest to the field of high energy density (HED) science [31]. As an example of a response of a solid let us consider a metal. In the optical regime, the interactions of femtosecond laser pulses with metals such as Ag and Au have been thoroughly studied, as reviewed by Rethfeld et al. [30]. Energy is first transferred from the photon to the electron system on the timescale of the incident pulse. The hot electron system then cools and equilibrates on timescales of 100 fs – 1 ps by interaction with the phonons of the initially cold lattice. This process is typically described by the so-called two-temperature model, based on different temperatures of the electronic system and the lattice [32, 33]. With time, the temperatures in the two thermodynamic reservoirs equilibrate, leading to the establishment of a FermiDirac electron distribution around the Fermi energy [34–36]. After deposition of large laser energy, the regular phonon modes of the lattice will become non-linear leading to severe atomic motion and diffusion that may culminate in melting. More recently, the initial optical studies of electronic-to-lattice heat transfer have been extended to ultrafast spin-lattice dynamics through the study of magnetic solids [37–40]. The two-temperature model is then replaced by a three-temperature model, where the third temperature is that of the spin reservoir. Strong perturbations of the electronic and magnetic structure, in particular the loss of the magnetization on femtosecond timescales, have been observed that has led to the concept of ultrafast “all optical magnetic switching”. Most of the optical studies were carried out at photon energies of ∼1.5–2 eV, pulse lengths of about 50 fs, and fluences of the order of 10 mJ/cm2 . Since such intensities of less than 1 mJ/cm2 /fs are at the lower end of possible XFEL intensities that extend to about 108 mJ/cm2 /fs or 1020 W/cm2 [41], it is clear that typical XFEL studies are bound to encounter strong non-linear effects of various types. In the following we will follow the time evolution of the processes in matter after being triggered by a femtosecond XFEL pulse, starting with the initial x-ray excitation of atoms. Since x-ray absorption has the largest cross section, the temporal evolution of the sample is triggered by the quasi-instantaneous x-ray absorption process during the incident pulse. In the process, part of the incident energy n ph ω contained in the x-ray pulse of length τ is transferred to the number of atoms Na = ρa V in the illuminated sample volume V = Ad, where d is the effective x-ray penetration depth. The fraction of atoms that are excited is determined by the XAS cross section σXAS which determines the penetration depth of the x-rays into the sample. While σXAS can vary by orders of magnitude for different atoms and photon energies, the atomic number density of solids varies relatively little and typically lies between 50 and 100 atoms/nm3 . Hence the penetration depth, typically expressed as one x-ray absorption length x defined as x = 1/(σXAS ρa ), mainly depends on σXAS . The initial transfer of energy mainly involves excitations of core electrons as illustrated in Fig. 1.27 on the attosecond timescale. As illustrated in Fig. 9.5, one needs to distinguish between resonant transitions to bound states and excitations into higher lying continuum states. In bound-state transitions, the excited electron is trapped and is not emitted as a photoelectron. Ignoring direct x-ray excitations of valence electrons, which have a lower cross section, changes of the valence states

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15 Non-linear Absorption and Scattering Processes in Solids

occupation will occur as an aftereffect of filling the created core hole on a timescale set be the core hole clock τ = / (see Fig. 12.4). Auger electrons created in the core hole decay are mainly responsible for electron reshuffling in the valence states through inelastic electron-electron scattering. The number of Auger electrons created in the core hole decay is typically larger than that of the spontaneously emitted xrays, and electron-electron scattering cross section is larger than photon-electron scattering cross sections. The kinetic energy of the Auger electrons of hundreds of eV is partially converted through inelastic electron-electron scattering and cascading into low energy electronic excitations of a few eV. While scattered electrons with kinetic energies above the surface potential barrier (work function) of 5 eV can escape into vacuum, most electrons (>90%) cascade down in less than 10 fs to energies below the work function barrier and spread over nanometers [42, 43]. We note that the scattering dynamics of electrons with kinetic energies in the 5–100 eV range is particularly complicated, and their equilibration is a topic of ongoing research [44]. In x-ray excitations to continuum states above the XAS “edge”, photoelectrons will be directly produced. They may scatter inelastically on their way out of a solid sample, leading to the well-known low energy electron tail in photoemission spectra. The photo electron scattering processes may therefore already lead to changes in the valence occupation before the core hole decays. The changes will be enhanced through the scattering of Auger electrons following the decay.

15.2.3 Energy Transfer to the Electronic System For both resonant and continuum state excitations, the electron scattering cascade leads to a hot electron distribution around the Fermi energy E F , caused by reshuffling electrons from below to above the Fermi level. The so-created electron/hole distribution around the Fermi level may significantly differ from the step-like lowtemperature distribution, where all states below E F are filled with electrons. This is illustrated in Fig. 15.1a where the filled states at low temperature are shown shaded in gray. In the figure we have assumed that at an elevated electronic temperature Te = T , the electron distribution is described by a Fermi-Dirac distribution function1 f (E) =

1

1 e(ε−EF )/kB Te

+1

.

(15.4)

We equate the Fermi level with the Fermi energy and denote them by E F . This ignores the difference between the Fermi energy and the chemical potential μ of a thermodynamic system which are strictly the same only at Te = 0. Since E F  kB Te , we have μ = E F to a very good approximation all the way to the melting point. This justifies the common practice of using the two quantities interchangeably for metals.

15.2 The Fundamental Damage Issue of XFEL Radiation

771

Electron/hole states near Fermi level

Convoluted difference

Difference in occupation

Fermi function = occupation

1.0

T (K) 300 2000 4000

(a)

0.8 0.6

filled electron states at T=0

0.4 0.2 0.0 0.4

T=4000-300 T=2000-300

(b)

0.2 0.0

holes excited electrons

-0.2 -0.4 0.15 0.10

T=4000-300 T=2000-300

(c)

0.05 electrons

0.00

holes

-0.05

EF

-0.10

convoluted with 0.43eV FWHM Lorentzian

-0.15

-2

-1 Energy

0

1

2

from E F (eV)

Fig. 15.1 a Electron distribution at T = 0 K indicated by gray shading, and its change according to the Fermi-Dirac function (15.4) at elevated temperatures T , for three values, where T is the equilibrated temperature T = Te = Tp of the electron and phonon systems. b Difference between the T = 2000 K and T = 300 K Fermi-Dirac functions in (a) shown (blue), and between the T = 4000 K and T = 300 K functions (red), revealing the reshuffling of electrons to states above E F and creation of holes below E F due to electron conservation. c Curves in (b) convoluted with a Lorentzian of FWHM = 0.43 eV (Co L3 value). The intensity difference due to electron/hole redistribution would be observed in an XAS experiment with spectral resolution limited by

Here ε − E F is the energy relative to the Fermi level, taken as the zero of energy in Fig. 15.1, Te is the temperature of the electronic system, and kB = 8.62 × 10−5 eV/K is the Boltzmann constant. In principle, the electronic temperature Te can only be defined if the electronic system has equilibrated with itself through electron-electron collisions and has the form of (15.4). After a sudden excitation of the electronic system, equilibration involves many collisions and occurs on timescales of about

772

15 Non-linear Absorption and Scattering Processes in Solids

10–100 fs [30, 34, 35, 45]. Figure 15.1a shows the distributions for three different temperatures. Figure 15.1b illustrates the change in distribution of electrons, corresponding to the difference of the T = 2000 K and T = 300 K function in (a), shown in blue, and between the T = 4000 K and T = 300 K functions (red). The change in temperature T leads to electron excitations to states above E F leaving behind hole states below E F due to conservation of electrons. In (c) we show the curves in (b) convoluted with a Lorentzian of FWHM = 0.43 eV (Co L3 value). The shown change in distribution would be measured in an XAS experiment, whose instrumental resolution is much less than the intrinsic lifetime width (see Fig. 15.3). For an incident number of photons n 0 , the absorbed number of photons n abs changes with depth x below the surface of a sample according to   n abs (x) = n 0 1 − e−σXAS ρa x .

(15.5)

The energy deposition hence has a large gradient. A film of thickness d, illuminated within an area A contains Na = ρa Ad atoms, and the average energy deposited per atom is given by Uat =

  F  n abs ω n 0 ω  = 1 − e−σXAS ρa d = 1 − e−σXAS ρa d . Na ρa Ad ρa d

(15.6)

During the short time of a femtosecond pulse, the deposited energy will be transferred almost exclusively to the electronic system whose electronic temperature is then defined by the change in Fermi-Dirac distribution.

15.2.3.1

From Energy Per Atom to Electronic Temperature

The Fermi-Dirac distribution function determines the temperature dependent occupation of electron and hole states around E F . In order to calculate the change in electronic temperature one needs to take into account the electronic density of states of a solid D(ε), which is typically listed in the literature in units of [states/atom/eV)] [46]. The zero of energy ε = 0 is taken to be at the bottom of the free electron-like valence states, and at zero temperature the states are filled up to the Fermi level E F which for metals is of order of 10 eV [46–48]. At Te = 0, the Fermi function is a step-like function so that the electron distribution is given by  n(ε) =

1 ε < EF 0 ε > EF

(15.7)

15.2 The Fundamental Damage Issue of XFEL Radiation

773

and the energy of the electronic system E0 = Ee (Te = 0) is ∞ E0 = n(ε) D(ε) dε.

(15.8)

0

For a free electron metal this works out to be E0 = 3Ne E F /5, where Ne is the total number of valence electrons per atom. We are interested in the change of energy of the electronic system per atom relative to its ground state value E0 . This arises from the rearrangement of electrons from below to above the Fermi level at finite temperature, expressed by the Fermi-Dirac distribution function. The problem was solved by Sommerfeld in 1928 [49] using a perturbation approach, but we use a simpler derivation. The change in energy per atom at an electronic temperature Te relative to Te = 0 is given by ∞ ε − EF π2 2 2 dε = D(E F ) k T . Ee  D(E F ) 2 (ε−E )/k T F B e + 1 e 6 B e

(15.9)

EF

We have approximated the density of states over the range of the Fermi-Dirac function by its value at E F and have used the fact that there is an equal number of electron and hole states which equally contribute to the energy change. This allows us to integrate from the Fermi level to infinity and simply multiply by 2. The so obtained value Ee is the energy associated with all valence electron excitations per atom. It is the temperature induced change of the ground state energy E0 = 3Ne E F /5. Our result agrees with the different derivation given in [47]. The specific heat of the electronic system per atom is obtained from (15.9) as ce =

δEe π2 = D(E F ) kB2 Te . δTe 3

(15.10)

The quadratic dependence on Te of the electronic energy associated with all excited electrons per atom given by (15.9) arises from the fact that the number of excited valence electrons per atom, Ne∗ , is given by Ne∗

∞ 1  D(E F ) 2 (ε−E )/k T dε = D(E F ) ln(4)kB Te . F B e + 1 e

(15.11)

EF

Since Ne∗ itself scales with kB Te , the energy normalized to each excited electron at an electronic temperature Te is given by Ee π2 = kB Te  1.19kB Te . Ne∗ 12 ln(2)

(15.12)

774

15 Non-linear Absorption and Scattering Processes in Solids

The small difference from the expected value kB Te per excited electron supports our approximation of pulling the density of states out of the integrals in (15.9) and (15.11). Equation (15.9) is derived by assuming a temperature-independent density of states and that the electronic temperature is not too high. It has recently been shown to hold remarkably well for transition metals up to temperatures close to the melting point [50]. It should be used with caution at larger electronic temperatures that can easily be created with XFEL x-rays. Nevertheless, by equating the deposited energy per atom with the change in electronic energy per atom, Uat = Ee , we obtain the temperature of the electronic system under the assumption that no energy is lost as  1 Te = kB

6 Uat . π 2 D(E F )

(15.13)

15.2.4 From Electronic to Lattice Temperature With increasing time, up to a picosecond or so, energy may be transferred between electrons and the lattice by electron-phonon collisions. Although electron-phonon and electron-electron collision times may be comparable [47], energy transfer from the hot electrons to the lattice will be slower due to the large mass difference of electrons and atoms. The electron-lattice equilibration is typically described phenomenologically by the so-called two-temperature model [32, 33]. It distinguishes the temperatures Te and Ta and specific heats ce and ca of the electronic system and atomic lattice. The equilibration of the systems is described by a time-dependent rate equation that links the two reservoirs through a coupling constant. Due to the difference in specific heats of the electron gas and the phonons [30], the equilibration may take picoseconds [30, 51, 52]. Historically, it was puzzling why the heat capacity per atom in the lattice described by the empirical law of Dulong and Petit2 could account for the specific heat of metals without needing to consider the contributions of the electrons per atom. Both atoms and electrons were expected to contribute equally based on the equipartition theorem. In its modern form, the Dulong-Petit law states that the heat capacity per atom has a constant value of (15.14) ca  3kB .

2

The law was suggested by P. L. Dulong and A. T. Petit in 1819 [53] as, “The atoms of all simple bodies have exactly the same capacity for heat.” They deduced from experimental data for 13 elemental solids a constant value of the heat capacity per weight multiplied by the relative atomic weight of the elements.

15.2 The Fundamental Damage Issue of XFEL Radiation

775

This relation can be derived in the Einstein and Debye models for large lattice temperatures, and it is a rather good approximation for transition metals at room temperature. The puzzle about the missing contribution of the electrons is naturally resolved by Fermi-Dirac statistics. It shows that only a small fraction of the electrons near E F can be thermally excited, while most of the energy of the electronic system is buried in the Fermi sea, extending from the Fermi-level down by about 10 eV to the bottom of the valence band. The electronic contribution of the excited electrons to the specific heat of the solid is very small at room temperature. This is directly seen by comparing the atomic specific heat ca per atom to the electronic specific heat ce per atom. They can be linked by comparing (15.10) and (15.14) as ce  3kB D(E F ) kB Te . 

(15.15)

ca

Since in metals D(E F )  1 atom−1 eV−1 [46] and at room temperature kB Te  0.025 eV, we have D(E F ) kB Te  0.025/atom. For this reason one typically finds ca + ce  ca . This is not valid at low temperatures of order 10 K, where the DulongPetit law does not hold and one may indeed detect the temperature dependent electronic contribution. After absorption of an intense x-ray pulse, the electronic temperature may be initially of order of kB Te  1−10 eV per atom. In the absence of heat loss, one may estimate the equilibration temperature with the lattice by using the constant DulongPetit value for the specific heat per atom. The electronic temperature then decreases by a factor of 3, and the lattice heats up to the same temperature. The increase T in lattice temperature, which determines whether the sample melts, can then be related to the fluence by use of (15.9) according to T 

 F  Uat = 1 − e−σXAS ρa d . ca 3kB ρa d

(15.16)

15.2.5 Ablation Threshold Ablation thresholds for XFEL radiation have been studied by Hau-Riege [54], Yumoto et al. [55], and Koyama et al. [56]. In Fig. 15.2 we show ablation patterns recorded as a function of x-ray fluence with 10 keV x-rays for 200 nm Pt layers deposited on SiO2 substrates [56]. Pt has an atomic number density of ρa = 65.1 atoms/nm3 , and at 10 keV an x-ray absorption cross section σXAS = 0.035 Mb/atom = 3.5 × 10−6 nm2 /atom [57]. The corresponding x-ray absorption length of 4.4 µm is much larger than the 200 nm thick film, and at the Pt ablation threshold F = 0.023 µJ/µm2 (see Fig. 15.2g), the energy absorbed per atom is obtained with (15.6) as Uat  0.5 eV/atom. According

776

15 Non-linear Absorption and Scattering Processes in Solids Ablation threshold for 200nm Pt on SiO2 substrate

(g)

Fluence

5

( J/ m2 = 10 mJ/cm 2 )

Fig. 15.2 Fluence-dependent ablation of Pt metal [56]. a–c SEM images of x-ray induced ablation craters for 200 nm of Pt on SiO2 substrates. d–f Cross-sectional profiles of the craters. The dashed line indicates the position of the Pt/SiO2 interface. g Imprint areas and ablation thresholds as a function of fluence for the Pt layer and SiO2 underneath

to (15.16) the rise in lattice temperature is given by T = 1930 K, and adding the initial room temparature of 300 K we obtain an ablation threshold temperature of 2230 K which compares to the melting temperature of 2041 K. These numbers are in reasonable agreement and support the simple theory outlined above.

15.3 Fluence-Dependent Changes of XAS Spectra

777

15.2.6 Summary We can summarize the evolution of the temperature in an x-ray excited solid as follows. After interaction with an intense x-ray pulse, the energy absorbed from the pulse per atom first ends up in the electronic system. After it has equilibrated within itself over a period of 10–100 fs, one can define an electronic temperature Te given by (15.13). The change of Te from absolute zero is described by the Fermi-Dirac distribution function, which expresses the change in electron and hole population around the Fermi level, E F . The population of electrons located in the deep Fermi sea well below E F remains unchanged. On the 100–1000 fs timescale the electronic system equilibrates with the lattice until they both have the same temperature. According to the DulongPetit law one finds as a rule of thumb an equilibrated temperature that is lower by a factor of 3 than the original temperature of the electronic system. This behavior is supported by experimental results for transition metals [34, 36].

15.3 Fluence-Dependent Changes of XAS Spectra In this section we illustrate two fundamental non-linear effects that are observed with increase of the incident intensity.

15.3.1 Redistribution of Valence Electrons As an example of a somewhat mild non-linear effect that appears at pulse fluences near the melting threshold we discuss the modification of the XAS L3 resonance of Co in a Co/Pd multilayer. Such multilayers have been extensively studied because they can be prepared with a stable magnetization direction along the surface normal, and have therefore served as a prototype system for exploring both modification of the valence charge by use of linear polarization and the valence magnetization by use of circular polarization. Here we only consider changes in the valence charge by use of spectra recorded with incident linear polarization. Intensity induced changes are illustrated in Fig. 15.3 for the 3d valence states of Co atoms in Co metal layers sandwiched between Pd [5, 6].

778

15 Non-linear Absorption and Scattering Processes in Solids

Optical pump, x-ray probe

X-ray pulses only

XAS intensity

1

(a)

0.8 0.6

Co L3 Co/Pd multilayer

40fs XFEL pulses 0.01 mJ/cm2 43 mJ/cm2

(c)

60fs optical pump 35 mJ/cm2 unpumped pumped

50fs XFEL probe . The atomic sheet enhancement factor Gcoh is then replaced by Geff as given by (15.40).

15.5.3 Sum Rule for Non-linear Film Transmission Our result for the NL transmission through a film of finite thickness d given by (15.37) with (15.40) may also be stated in terms of a sum rule which is the film version of the single atom sum rule discussed in Sect. 14.8 and illustrated in Fig. 14.15. By lumping NL , we can write (15.37) as all NL contributions into a NL XAS cross section σXAS 

Itrans

  ♦ ♦ ∞ = I0 exp − σXAS − 2 σXAS ρ

ρa d .   22

(15.42)

NL σXAS

NL corresponds to As for the atomic case, the so-defined NL XAS cross-section σXAS twice the NL REXS forward scattering cross section, i.e. NL NL = 2 σREXS . σXAS

(15.43)

This follows from the equivalence of the up and down rates in a two-level atom for ∞ ∞ = ρ11 expressed by (14.33) and (14.36). ρ22 As for the single atom case discussed in Sect. 14.8.1, we may therefore also divide the NL contribution into two equal parts according to ♦ ♦ stim stim + 2IREXS = I0 − (IXAS − IREXS ) + I stim Itrans = I0 − IXAS 



REXS  IXAS

IREXS

(15.44)

15 Non-linear Absorption and Scattering Processes in Solids

Fig. 15.9 Transmitted x-ray probability Itrans /I0 (green), given by (15.47), through a d = 20-nm-thick Co film for L3 -edge excitation as a function of the incident intensity I0 . Also shown is its sum rule decomposition according to (15.45) into an XAS loss intensity, IXAS (black), and stimulated REXS intensity IREXS (red)

20 nm Co film, L3 resonant excitation

1.0 Intensities rel. to I0

794

0.8

IXAS

0.6 0.4 Itrans 0.2 0.0

IREXS 1

10

100

1000 2

Incident intensity I0 (mJ/cm /fs)

The so-defined sum rule contributions identified on the right side of (15.44) or Itrans = I0 − IXAS + IREXS

(15.45)

are illustrated in Fig. 15.9 for resonant L3 -edge excitation of a 20-nm-thick Co metal film. The decomposition into XAS and REXS parts according to (15.45) is artificial, but it illustrates the existence of a sum rule in the BR theory due to the balance of up and down transitions at high intensity, leading to saturation. In the KHD perturbation theory, where saturation is not included, one typically assumes that the XAS contribution remains constant at the spontaneous value as a function of intensity and only stimulated forward scattering changes. The change in transmission would then only be half of that predicted by the BR theory. In the following we shall for simplicity denote all NL contributions as a NL change of absorption, so that the total XAS cross section is defined as ♦ tot NL = σXAS − σXAS . σXAS

15.5.3.1

(15.46)

The Generalized Beer-Lambert Law in the Presence of Stimulation

In the presence of stimulation, the conventional Beer-Lambert law for a film of arbitrary thickness d given by (15.37) may also be written in the terms of modified scattering lengths or optical parameters. The generalized Beer-Lambert law which includes stimulated forward scattering is then given by the modified spontaneous version of (15.24) as

15.5 Non-linear Transmission Through a Film of Finite Thickness

Itrans = I0 e

  ♦ NL ρa d − σXAS −σXAS

= I0 e−2λ( f



 − f NL )ρa d

= I0 e−2(β−βNL )kd .

795

(15.47)

The first formulation on the right side of (15.47) expresses the transmitted intensity ♦ given by (14.18) as through the change in the spontaneous atomic cross section σXAS ♦ = σXAS

λ2 x ( /2)2 . π (ω − E0 )2 + ( /2)2

(15.48)

The stimulated part is given by use of (15.40) as ♦ ♦ NL ∞ σXAS = σXAS 2[ ρ22 ]film  σXAS

I0 deff A , B + I0 deff A

(15.49)

where A and B are given by (15.30). The second formulation in (15.47) involves the imaginary part of the spontaneous scattering length, given by the quantum mechanical expressions of Sect. 13.4.1 as f  =

λ x

2 /4 . 2π (ω−E0 )2 + 2 /4

(15.50)

The NL contribution is given by  ∞ = f  2[ ρ22 ]film  f  f NL

I0 deff A . B + I0 deff A

(15.51)

 For the resonant case, the corresponding real parts, f  and f NL , are linked to the   imaginary parts f and f NL of the scattering length according to (13.19)

f =

2(ω−E0 )  2(ω−E0 )   = f , and f NL f NL ,



(15.52)

where the non-resonant Thomson contribution r0 Z to f  has been neglected. The third expression on the right side of (15.47) utilizes the link between the atomic scattering lengths and optical parameters derived in Sect. 7.5.5. The spontaneous optical parameter β is given by the quantum expression β=

λ2 ρa  ( /2)2 λ2 ρa λ x = f , 2 2 2π 2π (ω−E0 ) +( /2) 2π

(15.53)

and the NL part is ∞ ρ22 ]film  β βNL = β 2[

I0 deff A . B + I0 deff A

(15.54)

796

15 Non-linear Absorption and Scattering Processes in Solids

The corresponding real parts, δ and δNL are given by neglect of the Thomson contribution (see (7.52) and (7.53)) as δ=

15.6

2(ω−E0 ) 2(ω−E0 ) β, and δNL = βNL .



(15.55)

Polarization and Time-Dependent NL Transmission

Here we extend the polarization-averaged NL response in the long time limit given in Sect. 15.5.3.1 to include the polarization and time dependence.

15.6.1 The Polarization Dependent Generalized Beer-Lambert Law The polarization dependent Beer-Lambert law in the presence of stimulation is given by adding the polarization label p to the polarization averaged expressions in Sect. 15.5.3.1. In the process we can follow the polarization dependent treatments previously done semi-classically in Sect. 7.11.1 and for the KHD theory in Chaps. 10–13. The key polarization dependent parameter is the dipole transition matrix elep ment, represented by the dipolar transition width x , where the polarization averaged expression for a transition between two state |a ↔ |b given by (14.1) or 8π 2 αf ω |b|r ·  p |a|2 λ2

(15.56)

8π 2 αf ω |b|r ·  p |a|2 . λ2

(15.57)

x = is replaced by (10.40) or4

xp =

∞ The polarization averaged expression for [ ρ22 ]film given by (15.40) then includes a polarization dependent Rabi energy, defined as a polarization dependent interaction energy by generalization of (14.11). This leads to the polarization dependent version of (14.88) 3  p 2 pk p p λ p

= n xp . (15.58)

xp = n pk

R = I 0 2 2π c 2π 2 x

4

Also see the REXS formulation expressed by (13.21).

15.6 Polarization and Time-Dependent NL Transmission

797

For a film, Geff also becomes polarization dependent through the effective thickness deff given by (15.35) which now depends on the polarization dependent spon♦ ] p according to taneous XAS cross section [σXAS ♦

1 − e−[σXAS ]

p

deff 

♦ [σXAS ]p

p

ρa d

ρa

,

(15.59)

where (see (10.39)) ♦ [σXAS ]p =

p

λ2 x ( /2)2 . π (ω − E0 )2 + ( /2)2

(15.60)

We then have p

p

Geff =

λ2 ρa deff . 4π

(15.61)

With (15.58) we obtain the effective upper state population for a film as ∞ [ ρ22 ]film  p

p

p

=

p

Geff ( R )2 /4 p p (ω−E0 )2 + 2 /4 + Geff ( R )2 /2 p

p

I0 deff x ρa λ5 /(32π 3 c) . p p p (ω−E0 )2 + 2 /4 + I0 deff x ρa λ5 /(16π 3 c)

(15.62)

The Beer-Lambert law given by (15.47) becomes the following polarization ( p) dependent version p Itrans

=

 p  ♦ NL p ρa d p − σXAS −[σXAS ] I0 e

= I0 e−2λ( f p

 p

− f NL )ρa d p

= I0 e−2(β p

p

−βNL )kd p

.

(15.63) This shows that all expressions in previous chapters that describe the polarization dependent spontaneous transmission or diffraction response of a film can be extended to include the NL response by adding the respective NL contributions with a minus sign. The opposite signs lead to a cancelation of the spontaneous and stimulated contributions at high intensity, called self-induced x-ray transparency. The cross sections, scattering lengths, and optical parameters entering into (15.63) are explicitly given by the following expressions. The spontaneous cross section ♦ ] p is given by (15.60) and the NL contribution by [σXAS 

NL σXAS

p

  p   p ♦ ♦ ∞ p = σXAS 2ρ

22  σXAS film

By generalization of (15.30) we have

p

p

I0 deff A p . p p B + I0 deff A p

(15.64)

798

15 Non-linear Absorption and Scattering Processes in Solids p

x ρa λ5 , and B = (ω−E0 )2 + ( /2)2 . 16π 3 c

Ap =

(15.65)

The polarization dependent scattering length expressions are p

λ x

2 /4 2π (ω−E0 )2 + 2 /4

f  = p

(15.66)

and  ∞ = f  2[ ρ22 ]film  f  f NL p

p

p

p

p

p

I0 deff A p . p p B + I0 deff A p

(15.67)

Finally, the polarization dependent optical parameters are expressed through p

βp =

λ2 ρa λ x ( /2)2 2π 2π (ω−E0 )2 +( /2)2

(15.68)

and ∞ βNL = β p 2[ ρ22 ]film  β p p

p

p

p

I0 deff A p . p p B + I0 deff A p

(15.69)

15.6.2 Non-linear Polarization Dependent Transmission by the Magnetic 3d Metals As previously discussed in Sect. 10.5.5, XAS lineshapes for solids are often significantly broadened beyond the natural decay linewidths listed in Table 10.2, and the cross sections are proportionately reduced as illustrated in Fig. 10.17. For convenience we have listed typical polarization dependent parameters for the L3 resonances in Fe, Co, and Ni obtained from experiment in Table 15.1. The listed peak cross sections are those given in [14], and the linear polarization ones are the same as the polarization averaged ones plotted in Fig. 10.17 [see (11.29)]. In the following model calculations we will utilize the polarization dependent Co L3 response parameters of Table 15.1. In analogy to Fig. 10.17b, we show in Fig. 15.10 the polarization dependent the♦ ] p for a magnetically oriented oretical and experimental Co L3 cross sections [σXAS sample and x-ray propagation along the magnetization direction. The theoretical cross sections in (a) shown in blue have the natural Lorentzian FWHM = 430 meV, while the experimental cross sections in (b) (black lines) are compared to fits with Voigt profiles (red lines) consisting of a convolution of the natural Lorentzian lineshapes with a Gaussian of 1.4 eV FWHM. The latter accounts for band structure

Absorption cross section (Mb)

15.6 Polarization and Time-Dependent NL Transmission Co metal L3 -edge magnetic dichroism 8 (b)

(a) 20

799

theory

6

+

15

convoluted theory

experiment

+

+

0

0

0

-

-

-

4

10

2

5 0 775

776

777

778

779

780

0 781 775

776

777

778

779

780

781

Photon energy (eV)

Fig. 15.10 a Theoretical polarization dependent spontaneous Co L3 XAS cross sections σ p (blue lines) of Lorentzian lineshape as discussed in the text. b Comparison of the spontaneous experimen♦ ] p (black lines) and the theoretical cross sections in (a) convoluted tal cross sections σ p = [σXAS with a Gaussian of 1.4 eV FWHM (red lines). The corresponding blue Lorentzian profiles in (a) and red Voigt ones in (b) have the same areas

broadened d valence states into which the 2 p3/2 core electrons are excited. The areas under the blue curves in (a) and red curves in (b) are preserved. We are now interested in the change of the spontaneous cross sections, given by the red Voigt profiles in Fig. 15.10b, upon stimulation. The total polarization dependent NL cross sections for a 20-nm-thick Co film in the Beer-Lambert law (15.63) is given in accordance with (15.46) by  p   ♦ tot p NL p ] = σXAS − σXAS . [σXAS

(15.70)

The NL part given by (15.64) depends on the film thickness d which enters through p the polarization dependent effective thickness deff given by (15.59). As an example p we have listed in Table 15.2 the values for deff for a typical Co metal film thickness of p ♦ ] p in Table 15.1. d = 20 nm using the peak experimental cross sections σXAS = [σXAS p Also listed are the effective lateral coherent factors Geff given by (15.61) to illustrate the significant enhancement arising from coherent stimulated forward scattering. The total polarization dependent NL cross sections (15.70) for a 20-nm-thick Co film are shown in Fig. 15.11a for three values of I0 . They were obtained by p

Table 15.2 Polarization dependent effective thicknesses deff for a nominal Co metal film thickness of d = 20 nm, calculated according to (15.59) + − + − 0 (nm) 0 deff (nm) deff deff [nm] Geff Geff Geff 10.7

12.0

13.5 p

195 p

219

Also given are the values for Geff = λ2 ρa deff /(4π ) according to (15.61)

247

800

15 Non-linear Absorption and Scattering Processes in Solids

NL dichroic XAS spectra and peak intensity dependence

p

XAS

(Mb)

8

mJ

(a)

6 crosssection 4

I0 cm2 fs 0.1 50 200

polariz. p

(b)

+ 0

-

2

p Itrans I0

p

0 1.0 (c) 0.8 trans-

(d)

mission

0.6 0.4 0.2 775 776 777 778 779 780 781 0.1 Photon energy (eV)

1

10

102

103

104

2

Incident intensity I0 (mJ/cm /fs)

Fig. 15.11 a NL change of the polarization dependent XAS cross sections, indicated by dotted ( p = +), solid ( p = 0) and dashed ( p = −) lines, for Co L3 excitation for three incident intensities. The red curves for I0 = 0.1 mJ/cm2 /fs are indistinguishable from the corresponding red curves in Fig. 15.10b, representing the spontaneous response. b Change of the resonant (ω = 778 eV) cross section values in (a) as a function of incident intensity. The color coded vertical lines in (b) p p correspond to the three I0 values in (a). c NL transmission spectra Itrans /I0 according to the NL Beer-Lambert law (15.63) and d the corresponding dependence of the peak transmission value on the incident intensity in analogy to (a) and (b)

assuming that the spontaneous Voigt spectral shapes are preserved upon stimulation. Figure 15.11b shows the change of the resonant (ω = 778 eV) cross-section values in (a) as a function of incident intensity, where the color coded vertical lines correspond to the three I0 values in (a). In analogy to Figs. 15.11a, b we show in (c) and (d) the corresponding NL transp p mission spectra Itrans /I0 according to (15.63) and the dependence of the peak transmission value on the incident intensity. With increasing intensity the transmission becomes unity and the film becomes transparent. We note that our calculations were carried out with the assumption that the electronic populations have reached an equi∞ p ]film in (15.62). In the following we shall librium, i.e. t → ∞, as expressed by [ ρ22 explore how the cross sections and dichroic transmission of the same film changes for pulses of finite coherence time width τ , typically encountered in experiments.

15.6.3 Dependence on X-Ray Pulse Coherence Time We have seen in Sect. 14.1 that analytical solutions of the Bloch equation only exist in special cases, like the case of exact resonance and long time. In order to deal

15.6 Polarization and Time-Dependent NL Transmission

801

with time-dependent and non-resonant situations, the Bloch equations need to be solved numerically. For the film case, one then needs to account for the coherent p enhancement factor Geff . This is done by redefining the atomic Rabi frequency WR = p

R / in the Bloch equations (14.29) and (14.30) according to p

p





p

p

[WR ]film = WR Geff  WR

p

λ2 ρa deff , 4π

(15.71)

where the right side expression is obtained by use of (15.61). The analytical expres∞ p ]film is then replaced by the numerically calculated sion in the long time limit [ ρ22 p time-dependent excited state population [ ρ22t ]film . With these substitutions, our thin film expressions derived above are extended to include both the polarization and time dependence. In the following, we consider how the effective upper state population in a film changes for a SASE pulse containing coherent spikes of flat-top duration times up to τ . The temporal evolution of the population in a film is similar to that for the single atom case and we can conveniently use the results of Sect. 14.5.3. In analogy to (14.52), an integration up to a coherence time τ gives the effective upper population in a film as   p p p p 24 ωR 1 − cos(ωR τ ) e−3 τ/(4) Geff ( R )2 /4 τ p  1− [ ρ22 ]film = 2 p p p p

/4 + Geff ( R )2 /2 τ ωR 16(ωR )2 + 9 2    p p 16(ωR )2 − 9 2 sin(ωR τ ) e−3 τ/(4)  − , (15.72) p p τ ωR 16(ωR )2 + 9 2 where the generalized Rabi frequency is given by  p ωR

=

p 2

R

− 2 /16 

.

(15.73)

With increasing coherence time τ → ∞, the terms in wavy brackets in (15.72) p ∞ p ∞ p ρ22 ]film =[ ρ11 ]film = approach unity, and for large intensity ( R  ) we then obtain [ 1/2. As an example, we show in Fig. 15.12 the intensity-dependent increase in the excited state population (15.72) for L3 resonant excitation of a 20 nm Co metal film for different values of τ . We assumed a peak XAS cross section (linear polarization) 0 = 6.25 Mb corresponding to x0 = 0.34 meV. of Co atoms of σXAS τ 0 ]film with I00 shown in Fig. 15.12 shows a shift with decreasing The increase of [ ρ22 τ to higher intensity. This is expected because the driving field has less time to establish a stable upper state population. The shift resembles that previously shown in Fig. 15.8a with decreasing film thickness d. For short τ , the populations shown in Fig. 15.12 also exhibit Rabi oscillations that gradually damp out with increasing intensity.

Fig. 15.12 Excited state τ ]0 , given by population [ ρ22 film (15.72), as a function of the linearly polarized incident intensity I00 for different values of τ . We assumed L3 resonant excitation ω = E0 in a 20-nm-thick Co metal film as discussed in the text for linearly polarized light

15 Non-linear Absorption and Scattering Processes in Solids Upper state population for different pulse lengths

0.6

Population

802

=

0.5

infinity

0.4

20 fs 10 fs

0.3

3 fs 1 fs 0.5 fs

0.2 0.1 0.0 0.1

1

10

10

2

10

3

10

4

10

5

I00 (mJ/cm /fs) 2

Figure 15.13 illustrates the strong changes of the polarization dependent transmitted intensities for L3 excitation of a 20 nm Co film with both the incident intensity p I0 = I0 ( p = ±, 0) and length τ of incident coherent flat-top pulses. The plotted relative transmitted intensities are given by the NL polarization dependent Beer-Lambert law (15.63). In (a) we illustrate the transmitted dichroic intensities for τ = ∞ as a function of incident intensity, previously shown in Fig. 15.11d. Figure 15.13b shows their dependence on the pulse coherence time τ for three values of I0 , chosen in size and color as those in Fig. 15.11. In Fig. 15.13c, d we plot the dichroic difference between plus and minus angular momentum directions in (a) and (b). This corresponds to the transmitted XMCD signal in analogy to (7.123) for the spontaneous case. Again, we find a strong dependence on both I0 and τ , with the largest changes relative to the spontaneous contrasts occurring at large intensities and when the coherence time τ becomes significantly longer than the Co L3 core hole lifetime of τ = 1.5 fs, indicated by a vertical gray line.

15.6.4 From Collective to Independent Atomic Response In Sect. 15.4.2 we discussed the meaning of the coherent enhancement factor Gcoh ∞ for an atomic sheet which enters into the effective excited state population ρ

22 . In later Sects. 15.5.1 and 15.6.1, the treatment was extended to films of finite thickness and inclusion of the polarization dependence, resulting in the expression (15.61) for p ∞ p ρ22 ]film given by (15.62). Geff and the associated effective excited state population [ The complete polarization dependent effective excited state population in a film p reduces to that of a single atom through Geff → 1 according to (also see Sect. 15.5.2),

15.7 X-Ray Transparency

803

NL transmitted dichroic intensities versus I0 and coherent pulse length 1.0 (a) (b) core hole p Itrans I0

p

=

mJ cm2 fs

I0

0.8 polariz. p

0.6

-

0.4

+

lifetime

200 50 0.1

0

0.2 0.20 (d)

+ Itrans - Itrans

(c)

0.15 0.10 0.05 0.00 0.1

1

102

10

10 3

2

104 0.1

Incident intensity I0 (mJ/cm /fs)

1

10

100

Coherent pulse length p

(fs) p

Fig. 15.13 a NL change of the polarization dependent transmitted intensities Itrans /I0 according to (15.63), indicated by dashed ( p = −), solid ( p = 0) and dotted ( p = +) curves, as a function of incident intensity. All curves are for τ = ∞ and L3 excitation in a 20 nm Co metal film. b NL p p change of Itrans /I0 as a function of coherent flat-top pulse lengths τ for three color coded I0 values, indicated by vertical lines in (a), corresponding to those in Fig. 15.11. The Co L3 core hole lifetime of τ = 1.5 fs is indicated by a vertical gray line. c, d show the dichroic difference or transmitted XMCD signal derived from (a) and (b) ∞ ]film = BR : [ ρ22 p

p

p

Geff ( R )2 /4

p p ( E )2 + 2 /4+ Geff ( R )2 /2

∞ −→ [ρ22 ]atom = p

p

( R )2 /4

p

( E )2 + 2 /4+( R )2 /2

(15.74) where E = ω−E0 is the detuning energy.

15.7 X-Ray Transparency In this section we discuss the quantitative description of x-ray transparency, introduced in Sect. 15.3.2 for the two cases distinguished in Fig. 15.4a. We start with the description of transparency for resonant excitations which naturally follows from the two-level BR theory and compare it to experimental results.

804

15 Non-linear Absorption and Scattering Processes in Solids

15.7.1 Resonant Case: Co Metal To simplify things, we consider transmission through a film that is macroscopically isotropic. It may contain oriented microscopic magnetic or charge domains, but when averaged over the total illuminated area, the orientations average out and the transmission in the forward direction is independent of polarization. The microscopic structure will still lead to polarization dependent transmission differences between the domains and create an out-of-beam diffraction pattern as discussed in Chap. 8. The NL transmission and diffraction has been studied experimentally by Chen et al. [13] and Higley et al. [7] using linearly polarized XFEL SASE pulses of 25 fs and 5 fs length. The originally quoted length of 2.5 fs in [13] was later determined to be 5 fs [7]. The two studies used the same experimental arrangement, shown previously in Fig. 1.30. The relevant part of the complete experimental arrangement and the structure of the 25 fs SASE pulse in the energy domain is illustrated in Fig. 15.14. The x-rays were incident along the normal of the Co/Pd multilayer films of 25 nm Co thickness. The films contained magnetic stripe domains as shown in Fig. 15.14a. Their opposite magnetization directions were oriented perpendicular to the film plane, parallel, and anti parallel to the beam direction. For incident linear polarization, there is no conventional XMCD dichoism effect in the forward direction (q = 0), of interest here. However, at finite momentum transfer q  = 0 there is a domain diffraction pattern whose interplay with the centrally transmitted intensity will be discussed in Sect. 15.9. As shown in Fig. 15.14a, the pulses were split by a mirror with a sharp edge [61] into two similar intensity pulses that came to a focus at a Si chip containing 100-nmthick silicon nitride membrane windows. Half of the pulses were transmitted through the Co/Pd magnetic multilayers deposited on top of the SiN, and their intensity was normalized by the other half transmitted through pure SiN. The images shown on the CCD correspond to horizontally dispersed spectra, with the upper one showing a central dip due to transmission loss by the Co/Pd film. The normalization is illustrated in Fig. 15.14b by comparison of the transmitted spectra of two halves of single pulses, recorded with the membranes, and Co/Pd multilayers removed from the x-ray paths. The measured change of transmission through the film as a function of incident intensity of the 25 fs pulses is shown in Fig. 15.15 by data points shown as filled blue circles connected by a blue line. As mentioned earlier, the transmission for our linear polarization is independent of the magnetic orientation of the domains and is entirely a charge effect. Also shown as small blue open circles is a statistical simulation taken from [13]. The simulation was based on a statistical temporal description of the incident SASE pulses [62] which then served as input to the numerical solution of the time-dependent optical Bloch equations including the coherent enhancement factor. Here we take a shortcut to account for the observed effects by utilizing the anaτ ]film , given by (15.72), where τ represents the upper limit lytical expression for [ ρ22 of the width of the coherent SASE spikes. Inserting Geff = λ2 ρa deff /(4π ), given by (15.40), where deff has the approximated form (15.41), we obtain

15.7 X-Ray Transparency

805

CCD

Co/Pd multilayer

(a)

Grating Blank SiN

SASE pulses

E

Beamsplitter

Intensity

(b) Structure of split 25 fs pulses through blank frames

772

774

776

778

780

782

Photon energy (eV) Fig. 15.14 a Simplified schematic of the experimental setup [7]. The x-ray beam is split into two components with an x-ray beam splitter. One of the beams passes through a blank SiN membrane, while the other passes through a membrane with a Co/Pd magnetic multilayer with stripe domains, as shown. The domains had opposite magnetic orientations perpendicular to the film plane. The beams emerging from the membranes in the forward direction are measured with a grating-based spectrometer. For linear polarization, there is no dichroism effect in transmission, and the existence of the domains is unimportant, as discussed in the text. b Examples of single-shot spectra for a 25 fs pulse, recorded when the membranes and Co/Pd multilayers were removed from the x-ray paths

806

15 Non-linear Absorption and Scattering Processes in Solids X-ray transmission through Co/Pd film

Rel. transmission

1.0 0.8

Chen et al. experiment statistical theory

0.6

BR analytical =0.7-3.0 fs

=1.8 fs

0.4 0.2 1

10

100

4

1000

10

2

Incident intensity I0 (mJ/cm /fs) 10

2

10

3

10

4

10

5

2

Incident fluence 25fs pulses (mJ/cm ) -3

10

-2

-1

10

10

1

10

Photons per atom in film Fig. 15.15 X-ray transmission by a Co film, excited with 25 fs SASE pulses centered at the L3 resonance, plotted as a function of incident intensity, fluence, or number of photons per atom. The experimental data shown as filled blue circles connected by a blue line were taken from [13]. The small blue points also taken from that reference are a simulation based on the statistical temporal description of the incident SASE pulses [62] and a numerical solution of the time-dependent optical Bloch equations including the coherent enhancement factor. The dashed red curve is the normalized

22 (τ ) given by (15.75) and τ = 1.8 fs. transmission given by Itrans /I0 according to (15.76) with ρ The solid red curve includes an average over τ by integrating (15.75) over the range 1.7 ≤ τ ≤ 3.0 fs

ρ

22 (τ ) =

I0 deff x ρa λ5 /(32π 3 c) ( /2)2 + I0 deff x ρa λ5 /(16π 3 c)   p p 24 ωR 1 − cos(ωR τ ) e−3 τ/(4)  × 1− p p τ ωR 16(ωR )2 + 9 2    p p 16(ωR )2 − 9 2 sin(ωR τ ) e−3 τ/(4)  − . p p τ ωR 16(ωR )2 + 9 2

(15.75)

The transmitted intensity is given by the generalized Beer-Lambert expression ♦

ρ22 (τ )] . Itrans = I0 e−σXAS ρa d[1−2

(15.76)

The dashed red curve in Fig. 15.15 was calculated with (15.76) by use of (15.75) with τ = 1.8 fs. To match the experimental spontaneous transmission value of 0.32 [13], we used d = 20 nm as the Co thickness instead of the nominal value of d = 25 nm. The statistical pulse description used in [13] instead yields a spread of the small blue circles in Fig. 15.15. The Rabi oscillations in the dashed red curve in Fig. 15.15 are then obscured. The solid red curve in Fig. 15.15 is obtained by smoothing out the

15.7 X-Ray Transparency

807

Rabi oscillations by integrating (15.75) over a range 1.7 ≤ τ ≤ 3.0 fs. It describes the measured data quite well. We have now quantitatively accounted for the x-ray transparency effect previously shown in Fig. 15.4c. The early onset of transparency is a resonance effect where absorption is compensated by stimulated emission in the forward direction. The question arises how our resonant description of transparency developed above differs from that for non-resonant excitation, previously illustrated for Al in Figs. 1.28 and 15.4b. We now consider this case.

15.7.2 Resonant Versus Non-resonant X-Ray Transparency One may understand the difference between induced transparency for resonant and continuum excitations in terms of different lifetimes of the final state in the XAS process. In both cases it consists of an excited electron and a core hole. In a simple picture, the total lifetime of the final state may be separated into a contribution from the core hole, τh , given by the core hole clock FWHM lifetime τh = τ (see (12.18)) and from the free electron-like band state denoted τe according to [63] 1 1 1 = + . τtot τ

τe

(15.77)

The corresponding FWHM transition widths are given by the uncertainty relations

= /τ and e = /τe so that (15.77) in energy space reads

tot = + e .

(15.78)

In a resonant process, the excited electron remains bound for a time τe > τ

and does not escape as a photoelectron. The total excited state lifetime in (15.77) is therefore determined by the shorter core hole clock lifetime and τtot  τ (or

tot  in (15.78)). The process is described by the optical Bloch equations that couple the lower and upper states by “coherences”, as discussed in Sect. 14.3.1. The resonant process is determined by the stimulated increase in Rabi energy R2 = n x . In an atom, saturation occurs when through stimulation n x → and in a solid when n x Geff → , as expressed by (15.74). The resonant process depends on the bandwidth of the incident photons, i.e. their coherence time. Transparency occurs when the coherent up-down process reaches an equilibrium balance between absorption and stimulated emission, so that ρ22 = ρ11 = 0.5.

808

15 Non-linear Absorption and Scattering Processes in Solids

In a non-resonant process, core electrons are excited into continuum states well above the Fermi level with a rather smooth density of states. The excited electron first resides in a free electron-like band state of the solid before it can escape into vacuum through the surface potential barrier as a photoelectron. The lifetime τe may be thought of as a band-specific “trapping” time τe . If this trapping time is shorter than the core hole lifetime, τe < τ , the total excited state lifetime in (15.77) is determined by τtot  τe (or tot  e in (15.78)). This is the opposite of the resonant case. For metals, τe has been found to be a few hundred attoseconds [64]. For example, for Cu free electron-like s − p band states that lie about 20 eV above the Fermi level, it was found that τe  309 ± 94 as or e = /τe = 2.13 ± 0.65 eV [44]. The non-resonant XAS cross section is approximately independent of the incident photon bandwidth for typical values up to 10 eV or so. This means that it is insensitive to the coherence time of the incident radiation, and the coherent effects incorporated into the optical Bloch equations are absent. The up-down processes can simply be described by the Einstein rate equations discussed in Sect. 14.9. For solids, there is no coherent enhancement factor since the atoms respond independently. In the following we give a simple description of x-ray transparency based on the solution of the Einstein rate equations, by accounting for the exponential attenuation of the incident intensity in the film. When the time-dependent upper state population in the Einstein theory given by (14.122) is integrated over the pulse length τp , we obtain     ⎧ ⎫ 2 +2 R2 )τp ( tot ⎨ 2 − 1

tot  ⎬ exp −

tot 

R  2  ρ22 (τp ) = 2 1+ , (15.79) ⎭

tot + 2 R2 ⎩

tot + 2 R2 τp where the effective width tot  e + is determined by the larger of the two widths. In practice, for pulse lengths larger than a few femtoseconds, the term in curly brackets in (15.79) becomes unity. The upper state population, ρ22 (τp ), does not depend on the incident bandwidth or coherence time of the radiation. For the non-resonant excitation case, the pulses can therefore be incoherent “brute force” SASE pulses and both the photons and atomic excitation processes are uncorrelated. This is the key difference to the resonant case which requires coherent up-down processes, i.e. REXS. By dropping the time-dependent term and expressing the Rabi energy in terms of the incident intensity by use of (14.88), we obtain (15.79) in the form NR = ρ22

I0 x λ3 /(2π 2 c) . 2

tot + I0 x λ3 /(π 2 c)

(15.80)

The normalized transmission T = Itrans /I0 for the non-resonant case is given by Non−resonant : T = e−σXAS ρa d (1−2ρ22 deff /d ) , ♦

NR

˜

(15.81)

15.7 X-Ray Transparency

809

where d˜eff is given by (15.41). This compares to the resonant expression (15.76) or ♦

ρ22 (τ )) Resonant : T = e−σXAS ρa d (1−2

(15.82)

which depends on the coherence time τ through the excited state population ρ

22 (τ ) given by (15.75). The two transmission expressions differ in the form of the NL contribution, but NR = 2 ρ22 (τ ) → 1 both can lead to transparency, i.e. T = 1. This requires that 2ρ22 ˜ and deff → d.

15.7.3 Non-resonant Transparency Above the Al L-Edge We already illustrated the observed transparency for non-resonant excitation of Al above its L3,2 threshold in Figs. 1.28a and 15.4b and gave a qualitative explanation. We now show that the experimental results can be quantitatively described by (15.81). We start with the L3,2 XAS spectrum of Al metal shown in Fig. 15.16a [65]. One of the issues with Al metal films is that they readily oxidize unless they are capped or kept in ultra-high (∼10−10 Torr) vacuum [66]. The spectrum shown in Fig. 15.16a measured in transmission [65] likely contains small contributions from oxidized surface layers but is mostly representative of Al metal. Similarly, the films used in the transparency study by Nagler et al. [8] also contained oxidized layers. This complicates the analysis. We therefore consider the NL transmission for a pure Al metal film first. The properties of Al metal and its x-ray response are relatively well understood. Al has an atomic density of ρa = 60.3 atoms/nm3 , and the XAS cross section at ω = ♦  4.5 Mb 92 eV, about 20 eV above the L3 (72.7 eV) and L2 (73.2 eV) edges, is σXAS (4.5×10−4 nm2 ) as shown in Fig. 15.16a. This gives a 1/e x-ray absorption length of x = 1/(ρa σXAS ) = 36.9 nm,. The calculated transmission value Tspon = 0.24 is larger than the measured low intensity (spontaneous) value of Tspon  0.175 shown ♦ in Fig. 15.16b. The latter would require σXAS ρa d in (15.81) to be 22% larger than expected for pure Al, most likely due to film oxidation. The adopted value for the L3,2 -shell fluorescence yield is Yf = 7.5 × 10−4 , and the total core hole lifetime width is = 40 meV [67, 68]. This corresponds to a dipole matrix element x = Yf = 30 µeV. A theoretical study by Almbladh et al. [69] gave

= 22 meV and x = 45 µeV. The other quantity of interest in non-resonant XAS is the total upper state width according to (15.78). One would expect a similar band trapping time of the photoelectron as for Cu, corresponding to e = /τe ∼ 2 eV (τe ∼ 300 as) [44]. Since e is considerably broader than the core hole width , the upper state population (15.80) is described by tot = e . Let us take a look at the temporal evolution of the transparency process. For nonresonant XAS, the temporal coherence time of the incident beam does not matter. Then the intensity and fluence (intensity × time) are simply related by the pulse

810

15 Non-linear Absorption and Scattering Processes in Solids

5

(a) Al L3,2 region

Rel. film transmission

Cross section (Mb)

0.8

4 3

h = 92eV

2 1 0

70

90 Photon energy (eV)

110

(b) Al film, 53nm 15 fs pulses

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.1

1

10

1000

100

Incident fluence in15 fs pulses (J/cm²) 2

3

4

10 10 10 10 Incident intensity (TW/cm² = mJ/cm² /fs) 0.1 1 100 10 Photons per atom in 15 fs pulse

Fig. 15.16 a XAS spectrum above the Al L3 (72.7 eV) and L2 (73.2 eV) edges adopted from [65]. We have put the measured XAS spectrum on an approximate cross-section scale [57]. b Experimental data points for the NL increase of a 53-nm-thick Al film at ω = 92 eV relative the spontaneous value [8], shown on different abscissa scales. The solid red curve is the theoretical prediction by (15.81) as discussed in the text. The dashed blue curve is taken from the original work [8]

length. At a certain central excitation energy, the XAS cross section into continuum states is approximately constant when averaged over a bandwidth up to about 10 eV or so. At first, the x-ray penetration depth is determined by the spontaneous value x = 1/(ρa σXAS ), or the related effective spontaneous depth deff defined in (15.35). For a 53-nm-thick Al film we have x = 37 nm and deff = 28 nm. With increasing intensity, the core hole production extends deeper into the film since the atomic layers in the effective penetration depth become increasingly transparent. Transparency propagates layer by layer until the whole film becomes transparent, as reflected by d˜eff → d according to (15.36). As pointed out earlier, it is of great importance that the cross section for multiple Lshell core hole production is negligibly small relative to single core hole production. Hence transmission starts to increase approximately when there is on average one photon per atom, and it saturates when there are sufficient photons per atom that the excitation probability per atom becomes unity. If the atoms have a cross-sectional area Aat , saturation occurs when there are n ph  A/σXAS photons per atom. The area of Al atoms seen by the incident photons is Aat = 6.5 × 10−2 nm2 which compares to σXAS  4.5×10−4 nm2 . So it takes on average the attempts of n ph  150 photons per atomic area to create a single L-shell core hole. This is in good accord with Fig. 15.16b (see the lowest scale).

15.7 X-Ray Transparency

811

The other crucial parameter is the refilling time of the created core holes, given by their lifetime. For Al τ = / has the adopted value τ = 16 fs [67, 68] which compares to a theoretical value of τ = 31 fs [69]. For complete transparency to occur, the fluence in the short pulse length of τp = 15 fs would have to be large enough that the whole core hole production process is finished before it is partially compensated by core hole refilling, i.e. τp τ . Since this is marginally fulfilled for the Al L-edge case, complete transparency is not expected. This is indeed seen from the experimental data points in Fig. 15.16b which saturate below unity. The solid red curve in Fig. 15.16b was calculated with (15.81) using the exper♦ imentally observed value of Tspon = e−σXAS ρa d = 0.175, corresponding to a 22% ♦ increase of the product of the nominal values σXAS  4.5 Mb, ρa = 60.3 atoms/nm3 NR and d = 53 nm. The excited state population ρ22 was evaluated according to (15.80) with λ = 13.5 nm, tot  e = 1.7 eV (τe = 390 as), and the theoretical dipole matrix element x = 45 µeV [69]. NR ˜ In the product 2ρ22 deff /d in (15.81) we used d˜eff = 43 nm. This means that only NR ˜ and d˜eff the fraction deff /d = 0.81 of the entire film becomes transparent. Since ρ22 enter equivalently in the theory, one may also attribute the incomplete transparency NR = 0.81 averaged over all atoms in to an unsaturated excited state population 2ρ22 the entire thickness d˜eff = d. The physical origin of incomplete transparency is that the pulse length of τp = 15 fs is only slightly shorter than the core hole clock time τ  16−31 fs [67, 69], so that the exited state population can already partially decay during the pulse. The dashed blue curve in Fig. 15.16b, taken from the original work [8], is nearly identical to our solid red curve. The measured transparency has also been treated by Hatada and Di Cicco [70, 71] in a rate equation model with similar results.

15.7.4 Non-linear Transparency Above the Fe K-Edge A related investigation of x-ray induced transparency was carried out by Yoneda et al. using excitations to continuum states above the K-edge of Fe metal [10]. The XAS spectrum is shown in the inset of Fig. 15.17 as a gray curve [72] with the excitation energy of 7130 eV (λ = 0.174 nm) indicated by an arrow. The XAS cross section ♦ = 43 Kb/atom which corresponds to (right ordinate in the inset) has a value of σXAS the stated x-ray absorption coefficient by Yoneda et al..5 The experiment was carried out by monitoring the intensity-dependent transmission through a d = 20-μm-thick Fe foil (atomic density ρa = 85 atoms/nm3 ). Since the foil thickness greatly exceeded the spontaneous x-ray absorption length of x = ♦ 1/(ρa σXAS ) = 2.7µm, the spontaneously transmitted intensity, Tspon = e−σXAS ρa d −4 = 6.7 × 10 , was very small. The experiment involved 7 fs SASE pulses of large bandwidth (∼40 eV) with remarkably high intensities up to 1020 W/cm2 , produced 5 Note that the XAS cross section in Fig. 15.17 contains a background contribution of about 7 Kb/atom which was subtracted in Fig. 10.13.

812

15 Non-linear Absorption and Scattering Processes in Solids

40 K

2,5

XES

XAS

30 20 10

XAS cross section (Kb)

XES intensity (arb. units)

Transm. change T(I0) / Tref

Fe metal K-shell excitation

Photon energy (eV)

2

3 5 10 104 10 10 10 Incident intensity (PW/cm² = J/cm² /fs)

1

6

10

2 5 6 3 104 10 10 10 10 2 Incident fluence in 7fs pulses (J/cm )

10

-2

-3 1 10 10-1 10 10 Number of photons per atom in 7fs pulse

-4

10

Fig. 15.17 Inset: Fe K-shell XAS (gray curve) and Kβ2,5 (blue curve) XES spectra [72]. Main figure: Intensity-dependent NL transmission of 7 fs pulses, normalized the spontaneous transmission at low intensity, of a 20 µm Fe metal foil at a photon energy of 7130 eV, about 20 eV above the Kedge at 7112 eV [10]. The filled red squares and error bars are the experimental data points, and the solid blue and green curves are simulations taken from [10]. The red curve is calculated according NR evaluated according to (15.80) with σ ♦ to (15.81), with ρ22 XAS = 43 Kb/atom, λ = 0.174 nm, x = 4.5 meV, and tot = 1.33 eV. The dashed red curve corresponds to d˜eff = deff = 2.7 μm in (15.81)

through exquisite focusing onto the sample. The high intensities were required due to the relatively small K-shell XAS cross section (see Fig. 7.14) and the short core hole lifetime of about 500 as (see below), which made it difficult to create core holes on a significant fraction of all atoms in the film during the 7 fs pulse. Experimental data points for the intensity-dependent transmission, normalized to the spontaneous transmission value at low intensity, are shown in Fig. 15.17 as filled red squares with associated error bars. The solid blue and green curves are simulations taken from the original work, corresponding to core hole lifetimes of τ = 500 as (  1.3 eV) (blue) and 2 fs (  0.33 eV) (green). The solid red curve in Fig. 15.17 was calculated according to (15.81) with the nom♦ = 43 Kb/atom, ρa = 85 atoms/nm3 , and d = 20 μm. The excited inal values σXAS NR given by (15.80) was calculated with λ = 0.174 nm and the state population ρ22 following linewidths. For the Fe K-shell, the core hole width is given in the literature as = 1.25 eV [68] or = 1.4 eV [73]. Since these values are comparable to the expected excited state “band trapping” width, we used a value tot = 1.33 eV.

15.8 Polarization Dependent NL Transmission at Resonance

813

The dipole transition width x was determined from the partial fluorescence yield of the β2,5 emission line, shown in blue in the inset of Fig. 15.17 [72]. This emission line corresponds to dipole transitions from Fe 4 p valence states directly into the 1s core hole and its dipole matrix element x is the same as for the reverse 1s → 4 p core to valence excitation probed in XAS. According to theory6 [72], the β2,5 strength is almost exactly 1% of the total fluorescence yield Yf = 0.340 [67]. This ratio is supported by the measured value for Ni metal [74]. The dipole matrix element is then given by x  0.01 Yf , which by use of = 1.33 eV yields x = 4.5 meV. The red curve in Fig. 15.17 is seen to follow the experimental data points better than the published blue and green curves. In particular, the NL effect begins at lower intensity and initially follows the dashed red curve. It is calculated with d˜eff = deff = 2.7 μm in (15.81), which for the thick Fe foil is nearly the same [see 15.35)] as the spontaneous x-ray penetration depth x = 2.7μm. This shows that the early increase in transmission is due to the first stage of transparency induced in the spontaneous penetration depth. The transparency then progresses deeper into the foil until it increases more strongly when there is a photon for every atom in the film, marked by the vertical red line.

15.8 Polarization Dependent NL Transmission at Resonance The following relatively short section finishes the treatment of the NL transmission through a film, excited at resonance, by considering the largest NL transmission effect and its polarization dependence for a magnetically aligned sample. It also serves as the basis for the next Sect. 15.9 which introduces the link between in-beam transmission (q = 0) and out-of-beam (q = 0) diffraction.

15.8.1 The Maximum NL Transmission Effect The largest NL effect in x-ray absorption occurs when the incident pulses are temporally coherent for a time considerably longer than the core hole clock time, as illustrated in Fig. 15.12, and are tuned to the resonance energy. For the example of resonant Co absorption with a core hole lifetime of τ = 1.5 fs, the coherence time, needs to be about 20 fs or more. This situation is encountered either for a single transform-limited pulse of length longer than 20 fs, or for a longer SASE pulse that is put through a monochromator. In the latter case, the bandwidth of the monochromator filters out coherent fractions of the longer pulse, accompanied by a loss of intensity. A long pulse may then contain several coherent subpulses whose centers 6

I would like to thank Craig Schwartz for the actual numbers.

814

15 Non-linear Absorption and Scattering Processes in Solids

are temporally shifted. In coherence dependent experiments, the subpulses act independently and just improve the statistics of the measured signal. In the following we shall consider incident pulses of different coherence lengths. When divided by the coherence time, the pulses are assumed to have the same intensity. We will furthermore include the polarization dependence of transmission through a film, by considering that it is magnetically oriented along the x-ray propagation direction, chosen along the normal of the film. We know form Sect. 7.11.3 that incident circularly polarized x-rays will show a difference in transmission, the well-known XMCD effect, while the transmission for linearly polarized x-rays will be the average of the two circularly polarized cases. Of interest here is the intensitydependent evolution of the XMCD effect.

15.8.2 Polarization Dependent Transmission For a film that is uniformly magnetized in the direction mˆ z , the transmitted intensity is given by the polarization dependent ( p = 0, ±) NL Beer-Lambert law given by (15.63) or p

p   ♦ NL p p − σXAS −[σXAS ] ρa d

Itrans = I0 e

= I0 e−2λ( f p

 p

− f NL )ρa d p

= I0 e−2(β p

p

−βNL )kd p

. (15.83)

The cross sections, scattering lengths, and optical parameters in (15.83) are given in Sect. 15.6.1. In the following we will for convenience and consistency use the optical parameter formulation of Sects. 7.11 and 8.7. In those sections the spontaneous isotropic charge response was expressed through the optical parameters β 0 and δ 0 ( p = 0, linear polarization) given by (7.118), and the magnetic response was included through a change of these parameters denoted β and δ as defined in (7.117). In order to directly see the additional NL contribution we write (15.83) in analogy to the spontaneous expression (7.120) in the total, spontaneous plus stimulated, form, p,mˆ

Itransz = I0 e−2[(β p

0

0 −βNL )+ p mˆ z ( β− βNL )]kd

(15.84) p

Here the spontaneous charge contribution, β p ( p = 0, ±), and its NL part βNL , which is in general time dependent, are given by (see (15.69) and (15.68)) p

βp =

λ3 ρa x ( /2)2 p p , and βNL (τ ) = β p 2 ρ22 (τ ), 4π 2 (ω−E0 )2 +( /2)2

(15.85)

where ρ

22 (τ ) is expressed by (15.75). The magnetic corrections β and βNL are given by

15.9 Competition Between NL Transmission and Diffraction

β =

− β+ − β− β + − βNL , and βNL = NL . 2 2

815

(15.86)

The XMCD contrast is defined according to (7.122) and for the magnetization and polarization directions aligned according to mˆ z ± = ±1 we have + − IXMCD = Itrans − Itrans .

(15.87)

The NL form of the spontaneous expression (7.123) then reads IXMCD = −2I0 e−2(β

0

0 −βNL )kd

sinh[2( β − βNL )kd].

(15.88)

To illustrate the polarization and coherence time-dependent transmission, we reconsider the case shown in Fig. 15.15, corresponding to polarization independent transmission and a short coherence time of τ = 1.8 fs that is comparable to the core hole clock time of τ = 1.5 fs. We now compare it to the case of a magnetically aligned sample illuminated by a longer coherent pulse of τ = 20 fs of the same intensity. This is done in Fig. 15.18. In Fig. 15.18a we illustrate the effect of coherence time τ by plotting the total spontaneous plus stimulated transmission for a magnetically aligned sample according to (15.84) with mˆ z ± = ±1. The coherence time enters through the excited state population in (15.85). We see that the NL threshold for τ = 20 fs (blue curves) occurs at a lower intensity than for τ = 1.8 fs (red curves). In particular, the solid red curve for p = 0 and τ = 1.8 fs is the same as the dashed red curve in Fig. 15.15. The change in transmission due to stimulation relative to the spontaneous case is plotted in Fig. 15.18b. The solid curves reflect the change in relative transmission 0 0 /Ispon for the two different coherence times. The dashed for linear polarization, Itot + curves are the corresponding normalized dichroic XMCD ratios given by (Itot − − + − Itot )/(Ispon − Ispon ). It is interesting that for τ = 20 fs the change in transmission is nearly polarization independent as seen from the nearly identical solid and dotdashed blue curves. At τ = 1.8 fs the corresponding solid and dot-dashed red curves show differences caused by the Rabi oscillations. In the following sections we will discuss the competition between the forward and out-of-beam scattered intensity in the presence of stimulation. We will show that an increase in transmission comes at the expense of the out-of-beam scattered or diffracted intensity, as a consequence of energy conservation.

15.9 Competition Between NL Transmission and Diffraction In diffraction terminology, the transmitted intensity corresponds to zero momentum transfer, q = 0. We are now interested in the intensity scattered out of the beam,

15 Non-linear Absorption and Scattering Processes in Solids

p

Norm. trans. and XMCD Rel. transmission Itot

816

1.0 (a) polariz. p

0.8

0

+

0.6

=1.8 fs = 20 fs

0.4 0.2 1.0 0.8 (b) 0.6 0.4 0.2 0.0 -0.2 1

Itot0 0 Ispon Itot+ - Itot +- Ispon Ispon -

10

100

10 4

1000 2

Incident intensity I0 (mJ/cm /fs) Fig. 15.18 a Polarization dependent ( p = 0, ±) total (spontaneous plus stimulated) transmission p p Itot = Itrans as a function of pulse intensity according to (15.84). The red curves are for SASE pulses with coherent regions up to τ = 1.8 fs lengths, while the blue curves are for pulses of the same intensity with coherent regions up to τ = 20 fs lengths. The solid red curve for p = 0 corresponds to the dashed red curve in Fig. 15.15. b Total intensities in (a), normalized by the spontaneous parts. 0 /I 0 , and the dashed curves are the dichroic The solid curves correspond to linear polarization, Itot spon XMCD ratios according to (15.88) as indicated

i.e. the diffracted intensity at q > 0. One may first think that the weaker diffracted intensity remains unchanged from the spontaneous value since it is not stimulated. This is, however, not the case because of the fundamental law of conservation of energy between all available channels. An increase of intensity at q = 0 therefore comes at the cost of that at q > 0. We start by considering the NL change of the most fundamental diffraction pattern that of a homogeneous film with random bond and spin orientations in a circular aperture. The spontaneous diffraction pattern for this case, which is closely related to the emission characteristics of a circular source, was derived in Sect. 4.4.3 and later used in Sect. 8.6.3 to demonstrate Babinet’s amazing theorem. We now revisit it to demonstrate another fascinating phenomenon.

15.9 Competition Between NL Transmission and Diffraction

817

15.9.1 The NL Airy Pattern of a Film in a Circular Aperture We have discussed multiple times the case of a circular aperture of area As = π R 2 that is coherently illuminated with a constant intensity across the aperture. In Sect. 4.4.3 it served to illustrate the Fourier optics definition of coherence, in Sect. 8.6.1 to define the diffraction limit, and in Sect. 5.5.5 to illustrate the change of the first (one photon) to second order (two photons) diffraction pattern. The conventional (first order) Airy diffraction pattern as a function of momentum transfer q = kρ/z 0 is given by (4.71) or

I

(1)

 2  As 2J1 (q R) 2 2J1 (q R) 2 (1) As = I0 .   λ2 z 02 qR qR λ2 z 02  

(1) P

(q, z 0 ) = I0(1) As s

(15.89)

A0 (q)

We have added a superscript “(1)” to specifically indicate that the intensity distribution corresponds to single photon counts. The first underbracket identifies the power Ps(1) emitted by the source, expressed as the peak intensity times the source area. The power is preserved on propagation to the detector plane which is seen by substituting q = kρ/z 0 and spatial integration over the detector plane in cylindrical coordinates according to Pd(1)

2π ∞ = 0

I (1) (ρ, z 0 ) ρ dρ = Ps(1) .

(15.90)

0

The second underbracket in (15.89) identifies the unitless normalized shape of the Airy pattern A0 (q). It may be written by expressing q in units of the momentum transfer q0 = 1.22π/R of the first Airy minimum, which defines the Rayleigh diffraction limit, according to  2J1 (1.22πq/q0 ) 2 . A0 (q) = 1.22πq/q0

(15.91)

It was previously plotted as a black curve in Fig. 8.12b. In the presence of a film with random homogeneous bond and spin distributions, the corresponding spontaneous diffraction pattern is simply the Airy pattern of the hole, attenuated according to the conventional Beer-Lambert law, given by (8.25) or (1) (q, z 0 ) Ispon

=

I0(1)

 A2s 2J1 (q R) 2 −2βkd e qR λ2 z 02  

Aspon (q)

(15.92)

818

15 Non-linear Absorption and Scattering Processes in Solids

also previously shown in Fig. 8.12b as a blue curve, corresponding to 2βkd = 1. While diffraction by a hole alone or spontaneous diffraction by a film in a hole can be described by spherical wave emission from each point or atom in the aperture plane, the famous Huygens–Fresnel principle, this description breaks down in the presence of stimulated atomic scattering. In this case, the atoms in the film no longer spontaneously scatter photons into random directions, which on average is well modeled by the spherical waves assumed in the Huygens–Fresnel principle. Rather, the stimulating photons imprint their incident wavevector direction (mode) on the photon created in the stimulated decay. At maximum stimulation, the film becomes transparent as shown in Fig. 15.15.

15.9.1.1

Cloned Biphotons and the Two-Photon Airy Pattern

When the resonantly scattered (REXS) intensity is completely dominated by stimulation, one may envision all atoms as source points of two identical photons, which have been referred to as cloned biphotons [25, 26]. These biphotons are naturally coincident and lead to coincident two-photon counts at all points in the detector plane. The two-photon diffraction pattern is the squared one-photon Airy distribution given by (5.58) or I

(2)

 1 As 2J1 (q R) 4 (q, z 0 ) = .   1−16/(3π 2 ) λ2 z 02 qR  

(2) P I0(2) As s

(15.93)

2.176

The power-conserving prefactor 1/(1−16/(3π 2 )) = 2.176 is slightly larger than 2 because the central peak of the Airy distribution also robs most of the power contained in the outer rings for the spontaneous case, which decreases from 16.2% to 0.2%. The total power is preserved on propagation to the detector plane which is seen by integrating over the detector plane in real space cylindrical coordinates according to Pd(2)

2π ∞ = 0

I (2) (ρ, z 0 ) ρ dρ = Ps(2) .

(15.94)

0

From a detection point of view, care has to be exercised with the cloned biphoton picture. As discussed in Sect. 5.6.1 one needs to distinguish a collective two-photon state given by (5.64) from its binomial two-photon substate identified in (5.66). The collective state contains an average number of two photons statistically distributed over a Poisson distribution shown in Fig. 5.11b, while the two-photon substate contains exactly two photons. These two-photon states differ in their degree of second order coherence as discussed in Sect. 5.6.3.1. The two-photon substate is not second order coherent, but it can be picked out by two-photon coincidence detection as demonstrated in [75]. In contrast, the collective two-photon state is second order

Relative diffracted i ntensity

15.9 Competition Between NL Transmission and Diffraction

(a)

1 -1

10

hole

2.0

film, spont. film, stim.

1.5

-2

819

(b)

10

0.61q0

-3

1.0

10

-4

10

0.84q0

0.5

-5

10

-6

10

0

0.5 1.0 1.5 2.0 2.5 3.0 3.5

0.0 -1.0

- 0.5

0

0.5

1.0

Momentum transfer rel. to first Airy node, q /q0

Fig. 15.19 a Airy diffraction patterns A0 (q) (black) according to (15.89), Aspon (q) (blue) defined in (15.92) and Astim (q) given by (15.95) (red) as a function of normalized momentum transfer q/q0 , where qo = 1.22π/R is the first node of the Airy pattern. We have assumed a circular hole of radius R = 725 nm and λ = 1.6 nm (Co L3 resonance) and for the blue curve a film attenuation by a factor of 3. b Enlarged central part of the patterns on a linear scale, revealing the reduction of the width and increase of the intensity of the stimulated pattern

coherent and its diffraction pattern is naturally recorded by use of a photon number integrating detector such as a CCD. This is discussed in more detail in Chap. 16. In the following we will not worry about the detection details but are interested only in the difference between the one-photon and two-photon Airy patterns. The two-photon diffraction pattern (15.93) corresponds to the completely stimulated pattern of a film emitting biphotons given by (2) (q, z 0 ) = I0(2) Istim

 A2s 1 2J1 (q R) 4 . qR λ2 z 02 1−16/(3π 2 )  

(15.95)

Astim (q)

Note that the one-photon intensities given by (15.89) and (15.92) and the two-photon intensity in (15.95) are all normalized to the incident number of photons so that I0(2) = 2I0(1) . In Fig. 15.19 we have compared the normalized patterns per photon for a circular hole A0 (q) (black, see (15.89)), a Co film in the hole Aspon (q) (blue, see (15.92)), and its completely stimulated pattern Astim (q) (red, see (15.95)). We used the parameters of the experiment reported in [297], λ = 1.6 nm (Co L3 resonance) and a circular aperture hole of R = 725 nm. The patterns are plotted as a function of momentum transfer q/q0 , where q0 = 1.22π/R is the first node of the Airy pattern. The conventional normalized Airy pattern of the hole shown as black curves in Fig. 15.19 contains 83.8% of the total power in the central disk. The stimulated pattern, given by (15.95) and shown in red, contains 99.8% of the power in the central disk, so that the outer Airy rings have negligible intensity. The spontaneous Airy pattern in the presence of a film given by (8.25) is shown in blue, assuming the

820

15 Non-linear Absorption and Scattering Processes in Solids

presence of a d = 20-nm-thick Co film with optical constant β = 0.0075. In this case, the intensity and power are reduced by a factor of 3 through absorption in the film. Figure 15.19b reveals the narrowing of the central Airy disk from the full width half maximum (FWHM) value 0.844 q0 for the hole (black) and spontaneous film (blue) patterns, to 0.606 q0 for the stimulated film (red).

15.9.2 Change of the Spontaneous to the Stimulated Pattern Our above discussion was restricted to the case of complete (saturation) stimulation. In order to derive the evolution with incident intensity from the blue spontaneous pattern in Fig. 15.19 to the red stimulated pattern, we describe the response of the film in terms of the spontaneous β and non-linear βNL optical parameters (see Sect. 15.6.1). The development of the pattern is obtained by considering that the total (spontaneous and stimulated) q-dependent diffracted power P in the detector plane has to be equal to the transmitted power (at q = 0) in the exit plane of the sample, so that [12] Ptrans = Fspon Pspon (q) + Fstim Pstim (q) .  

 





exit plane

spon. pattern

(15.96)

stim. pattern

Here Fspon and Fstim with Fspon + Fstim = 1 are suitable intensity-dependent but qindependent coefficients that determine the relative contributions of the q-dependent spontaneous and stimulated patterns. We can determine the coefficients Fspon and Fstim by integrating the spontaneous and stimulated patterns in cylindrical coordinates over q, or equivalently over the real space distance ρ = qz 0 /k from the optical axis in the detector plane, and with I0 = I0(1) = I0(2) /2 we obtain Ptrans = I0 As e−2[β−βNL ] kd 2π ∞ (1) Pspon = Ispon (ρ, d) ρ dρ = I0 As e−2βkd 0

0

2π ∞ Pstim = 0

(1) Istim (ρ, d) ρ dρ = I0 As ,

(15.97)

0

(1) (2) (1) (q, z 0 ) is given by (15.92) and Istim (q, z 0 ) = Istim (q, z 0 )/2 is the twowhere Ispon photon intensity (15.95) normalized per photon. By inserting these expressions into (15.96) we can determine the coefficients and obtain the total power-normalized Airy diffraction pattern as

15.9 Competition Between NL Transmission and Diffraction

Itot (q, z 0 ) =

821

1− e−2(β−βNL )kd (1) e2βNL kd − 1 (1) Ispon (q, z 0 ) + 2βkd I (q, z 0 ), −2βkd 1− e e − 1 stim  

 

Fspon

(15.98)

Fstim

where Fspon + Fstim = 1.

(15.99)

The coefficient Fspon reflects the relative decrease of the spontaneous pattern, and Fstim the relative increase of the stimulated pattern under the condition of power conservation. The NL response arises through the intensity-dependent optical parameter βNL given by (15.69) or I0 deff A βNL = β . (15.100) B + I0 deff A The total pattern is seen to reduce to the spontaneous limit for βNL = 0 and the stimulated limit for βNL = β, as required. In Fig. 15.20a we plot the intensity-dependent change of the Airy diffraction pattern Itot (q), given by (15.98) and normalized by I0 A2s /(λ2 z 02 ), for a resonantly excited 20-nm-thick Co film in a circular aperture. We used the L3 resonance value ♦ =6.25 Mb. β = 0.0072, corresponding to σXAS The q-dependent patterns in Fig. 15.20a are shown for four incident intensities I0 , assuming laterally coherent x-rays with a temporal coherence length of 20 fs. The patterns contain different contributions of the normalized (intensity independent) spontaneous pattern Aspon (q), defined in (15.92), and its square, the stimulated pattern Astim (q) defined in (15.95), weighted by the intensity-dependent coefficients Fspon and Fstim . They evolve from the blue pattern for I0 = 1 mJ/cm2 /fs which is nearly identical to the blue spontaneous one in Fig. 15.19, to the orange pattern for I0 = 3 × 103 mJ/cm2 /fs, which approaches the red (completely) stimulated pattern in Fig. 15.19. In (b) we show the enlarged central part of the patterns in (a) on a linear scale to more clearly see the stimulated increase. In Fig. 15.20c we show as a blue curve (left ordinate scale) the I0 -dependent decrease of the total outer Airy ring intensity integrated over all q. It is given by Fspon in (15.98). Superimposed as black-gray circles are the experimental data points by Wu et al. [12]. The intensity of the 50 fs pulses monochromatized SASE pulses used in the experiment was assumed to be contained in two coherent 20 fs subpulses to match experimental and theoretical coherence lengths. The experiment is discussed in more detail in Sect. 15.10.2. Also shown as a red curve (right ordinate) is the I0 dependence of the cental q = 0 Airy intensity. It is given by Itot (q = 0) in (15.98), normalized by I0 A2s /(λ2 z 02 ). Superimposed on the red curve as filled colored circles are the peak values of the four curves in (b) shown in corresponding colors.

822

15 Non-linear Absorption and Scattering Processes in Solids

15.9.3 X-Ray Soliton Model: Mode-Dependent Stimulation The change of the diffraction pattern of a film with a homogeneous charge distribution in a circular aperture shown in Fig. 15.20 may also be described by a model based on the development of a spatial light soliton in the forward direction (q = 0) through a non-linear propagation process through the thin film. In such a two-step model the circular aperture is first assumed to diffract the incident first order coherent beam into modes of different directions and intensities. In the second step the modes propagate non-linearly through the film to create a modified pattern as illustrated in Fig. 15.21. In the optical regime, the propagation of spatial light solitons [76–78] is typically described semi-classically by an intensity-dependent change of the refractive

(a)

1 mJ/cm2/fs 30 mJ/cm2/fs 100 mJ/cm2/fs 3000 mJ/cm2/fs

-2

10

2.0

(b)

1.5

-3

10

1.0

-4

10

0.5

-5

10

-6

10

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

-1.0

-0.5

Rel. outer Airy ring intensity

Rel. momentum transfer

/

2.0

(c)

1.5

0.6 0.4

1.0

0.2

0.5

0.0

0.0 0.1

1

10

100

0.5

1.0

0

1.0 0.8

0.0

Central Airy disk intensity

Relative Airy intensity

NL Airy ring (charge) diffraction intensity of a film in a circular aperture 1 10-1

1000 2

Incident intensity I0 (mJ/cm /fs)

Fig. 15.20 a Airy patterns versus momentum transfer q in units of the first node of the Airy pattern at q0 = 1.22π/R for a 20 nm Co film in a circular aperture for four values of I0 . The patterns were calculated with (15.98) and normalized by I0 A2s /(λ2 z 02 ). We assumed Co L3 resonant excitation (λ = 1.6 nm) by laterally coherent x-rays with a longitudinal coherence time of 20 fs. b Central part of the pattern around q = 0 on a linear intensity scale calculated with the same parameters. (c) The blue curve (left ordinate scale) is the I0 -dependent change of the total outer Airy ring intensity given by Fspon in (15.98). It is compared to the data (black-gray circles) measured by Wu et al. [12] assuming the same 20 fs coherence length. The red curve (right ordinate) is the I0 dependent change of the total Airy intensity Itot (q) at q = 0, given by (15.98) and normalized by I0 A2s /(λ2 z 02 ). The peak values of the four curves in (b) are superimposed on the red curve as filled circles of corresponding colors

15.9 Competition Between NL Transmission and Diffraction Au

823

reference hole

A= R 2

sample with magnetic domains

log intensity diffraction modes

Si3N4 membrane

km

k0

k0 k1 k2

Magnetic domain pattern

qm

Airy charge pattern

z0 Fig. 15.21 Typical sample structure used for coherent imaging experiments. The beam is transmitted and diffracted by a thin film containing domains with the beam size defined by a cylindrical Au aperture of area A = π R 2 . The sample structure is shown in detail in the inset. The diffracted beam is detected by a position sensitive detector at a distance z 0 from the sample. The Airy pattern due to the sample aperture and a schematic magnetic diffraction pattern determined by the average size of the domains is shown on the right, plotted on a logarithmic intensity scale that emphasizes the small outer modes. One may distinguish different diffraction modes by their wavevectors ki , as shown. The transmitted mode in the forward direction k0 has by far the largest intensity and is preferentially stimulated

index through the optical or AC Kerr effect [79]. The dependence of the refractive index of the medium on the intensity creates a refractive index change that mimics the transverse intensity pattern of the beam. This causes the natural linear beam expansion to become compensated by the non-linear self-focusing. The self-focusing effect depends on the shape of the intensity distribution in the material which modifies the refractive index. This may be expressed as a non-linear contribution that is proportional to the third order non-linear susceptibility [1]. For the spontaneous response case, the Airy diffraction pattern from the circular aperture, shown in black in Fig. 15.21 on a logarithmic scale, is attenuated by absorption in the film relative to an aperture without a film as previously illustrated in Fig. 15.19. At low incident intensities, the spontaneous charge response is independent of the location of the film which simply acts as an attenuator. For the stimulated case, the location of the film behind the aperture matters since the intensity decreases with distance from the aperture. The strongest non-linear effect will occur when the film is located just behind the circular aperture, and it is weakest when it is inserted right before the distant detector due to the spatial expansion of the beam. In general, the q-dependent stimulation process consists of two contributions. First, at a given q (mode) the intensity change is dependent on the intensity in this mode I (q). Second, there is intensity exchange between different q directions (modes), driven by optimization of total power conservation.

824

15 Non-linear Absorption and Scattering Processes in Solids

In the following we want to quantify the soliton model. Ignoring polarization labels to simplify notation, the incident and spontaneously diffracted intensity distributions are given by 

I0 − R 0. We know that the centrally transmitted intensity is enhanced by stimulation at the expense of the diffracted intensity, under the constraint of power conservation. We can therefore divide the

828

15 Non-linear Absorption and Scattering Processes in Solids

total intensity in the detector plane into these two contributions. We furthermore replace the direction dependent momentum transfer q > 0 by performing a cylindrical average around the central beam, so that Itot (q) = a Itrans (q  0) + b Idiff (q > 0),

(15.117)

where the coefficients a and b depend on the incident intensity and have to satisfy power conservation. The situation is similar to the division of the Airy pattern of an aperture into the central Airy disk intensity and an average over the outer ring intensity, discussed in Sect. 15.9.2. Because of Fstim + Fspon = 1 we can associate the coefficients a and b in (15.117) with a = Fstim and b = Fspon to obtain, Itot (q) = Fstim Itrans (q  0) + Fspon Idiff (q > 0).

(15.118)

The transmitted intensity is given by (15.84) which for linear polarization p = 0 is independent of mˆ z and given by Itrans (q  0) = I0 e−2(β−βNL )kd .

(15.119)

The diffracted intensity due to the magnetic domains is given by (15.116) Idiff (q > 0) = I0

A2eff −2[β−βNL ] kd e CNL |Dmˆ z(q)|2 , λ2 z 02

(15.120)

where all the q-dependent information is contained in Dmˆ z(q), which for a given q has been cylindrically averaged around the centrally transmitted intensity. We can then divide the total intensity in the detector plane into in-beam transmitted and out-of-beam diffracted components, and by use of (15.116) and demanding power conservation we obtain Itot (q, ω) = I0 e

−2(β−βNL )kd

 F (1−C ) + Fspon CNL  stim  NL

 central transmission

 A2eff 2 |Dmˆ z(q)| . λ2 z 02 

out-of-beam diffraction

(15.121) Through the explicit photon energy dependent notation Itot (q, ω) we indicate that the optical constants implicitly are strongly photon energy dependent. This is particularly important for diffractive imaging since β(ω) and δ(ω), and their dichroic and NL contributions have their maximum values at different energies as illustrated in Fig. 8.30. We first discuss the terms in (15.121) by assuming that the incident x-rays of narrow bandwidth are tuned to the XAS maximum, i.e. the maximum of β(ω). We will come back to the photon energy dependence in Sect. 15.10.3 when we discuss the case of broad bandwidth incident x-rays.

15.10 Polarization Dependent NL Diffraction

829

The NL transmission through the film is described by the modified exponential Beer-Lambert law by inclusion of βNL = β

I0 deff A . B + I0 deff A

(15.122)

We have βNL → 0 for small I0 and βNL → β for large I0 . The intensity-dependent coefficients balance the transmitted and diffracted contributions through their dependence on βNL according to the limits Fspon =

1− e−2(β−βNL )kd , 1− e−2βkd

βNL →0

=⇒ 1, and

βNL →β

=⇒ 0

(15.123)

and Fstim =

e2βNL kd − 1 , e2βkd − 1

βNL →0

=⇒ 0, and

βNL →β

=⇒ 1.

(15.124)

The diffracted pattern defined by the second underbracket in (15.121) is given by the Fourier transform of the binary domain pattern |Dmˆ z(q)|2 (see (15.115)). The factor A2eff /(λ2 z 02 ) describes the propagation from the illuminated sample area Aeff to the detector plane at distance z 0 . The strength (intensity) of the pattern depends on the incident intensity I0 through the product Fspon CNL . The magnetic domain response is expressed by CNL which depends on the dichroic spontaneous and NL parameters taken up to second order according to,7 & ' CNL = [ β − βNL ]2 + [ δ− δNL ]2 k 2 d 2 βNL →0

=⇒ Cspon = [( β)2 + ( δ)2 ]k 2 d 2 , and

βNL → β

=⇒

0,

(15.126)

where we have indicated the limits. Note that Cspon represents the strength of the spontaneous magnetic diffraction pattern which is small relative to the centrally transmitted intensity (of order 10%).8 We obtain the following spontaneous and completely stimulated limits of the intensity in the detector plane

7

The expression given for CNL in [12] differs in the definition of the sign of the NL term from that in [13] which is the same as used here. It reduces in second order of the optical constants to our expression according to CNL = cosh [2( β − βNL )kd] − cos [2( δ − δNL )kd]    2 ( β − βNL )2 +( δ − δNL )2 k 2 d 2 . 8

(15.125)

For example, in the Airy diffraction pattern, the outer rings contain 16.2% and the central disk 83.8% of the total intensity (see Sect. 15.9.1.1).

830

15 Non-linear Absorption and Scattering Processes in Solids

 spon Itot (q)

=

A2

Idiff (q > 0) = I0 e−2βkd λ2effz 2 [( β)2 +( δ)2 ]k 2 d 2 |Dmˆ z(q)|2 (15.127) & 0 ' Itrans (q = 0) = I0 e−2βkd 1−[( β)2 +( δ)2 ]k 2 d 2  I0 e−2βkd  stim Itot (q) =

Idiff (q > 0) = 0 . Itrans (q = 0) = I0

(15.128)

In the following we will consider two distinct NL diffraction cases for a sample with ferromagnetic domains, extending the spontaneous diffraction case treated in Sect. 8.7.3. Our theory developed above will be compared to experimental data obtained with pulses of coherence times longer than the core hole life time by Wu et al. [12] and pulses with comparable coherence and core hole life times by Chen et al. [13].

15.10.2 Narrow Bandwidth Resonant Case We first consider an incident beam of narrow bandwidth and long coherence time, tuned to the peak of an XAS resonance. This case was studied experimentally by Wu et al. [12] as illustrated in Fig. 15.23a. The same geometry was previously shown in Fig. 8.18 to illustrate the separation of magnetic and charge diffraction from magnetic domains in a Co/Pd film, excited at the Co L3 resonance. The experiment was conducted at LCLS using linearly polarized laterally coherent SASE pulses of nominal 50 fs length. The pulses were sent through a monochromator which defined the photon energy of 778 ± 0.1 eV, matching the Co L3 transition. They were then focused to 10µm, overfilling the 1.45µm circular Au aperture located before the samples. As shown in Fig. 15.23a the Au apertures contained five 100 nm reference holes that were completely drilled through and were used to measure the pulse fluences F incident onto the sample by use of a holography trick [80] (see Sect. 8.9.2). Simulations of the SASE pulses transmitted through the Gaussian monochromator transmission function of 0.2 eV FWHM yielded total output pulses containing on average 1.5 temporally coherent regions of about 20 fs width [12]. Since each coherent 20 fs subpulse acts independently in a coherent REXS diffraction process, the average experimental intensity per total pulse was divided by 1.5 in order to compare it to the theoretical coherent intensity I0 per pulse in units of [mJ/cm2 /20 fs]. The so-determined coherent intensity per 20 fs is then simply divided by 20 to obtain the coherent intensity in units of [mJ/cm2 /fs] that will be used below. The sample was a Co/Pd multilayer, containing 20 nm of Co, with a nanoscale magnetic worm domain structure as shown in Fig. 15.23a. The diffraction pattern was recorded with a position sensitive CCD detector, with the total accumulated charge per pixel forming the diffraction pattern. No temporal gating of the detector and no electronic coincidence circuit were employed.

15.10 Polarization Dependent NL Diffraction

(a)

831

Recorded pattern

Aperture pattern beam stop

Co/Pd film with magnetic domains in Au aperture with reference holes

Domain pattern

E 778 eV (c)

0.6 mJ/cm²/pulse

272 mJ/cm²/pulse

Experiment

Radial intensity average

1500

(b)

(d)

0.6 mJ/cm²/pulse

12

272 mJ/cm²/pulse

8

magnetic domain diffraction

4 0

0

2

4

0

Theory of Airy intensity (e)

6 8 10 0 2 4 Momentum transfer / 0

x10

6

8

10

Fig. 15.23 a Coherent imaging experiment of a magnetic Co/Pd film with magnetic worm domains in a circular Au aperture illuminated by laterally coherent monochromatized XFEL pulses (FWHM 0.2 eV bandpass) tuned to the Co L3 resonance at 778 eV [12]. A beam stop in front of the CCD detector eliminated the intense central part of the diffraction pattern. The linearly polarized incident radiation allows separation of the diffraction pattern from the Au aperture and the magnetic domains due to the polarization dependence of charge and magnetic scattering as shown. b Experimentally recorded diffraction pattern by averaging over many low-fluence shots (0.6 mJ/cm2 /pulse). c Pattern for a single high-fluence shot (272 mJ/cm2 /pulse). d Radial intensity average of the patterns in (b) and (c) for the two fluence values. e Simulation of the change in the Airy pattern of the aperture, assuming the presence of stimulated resonant scattering [25, 26]

832

15 Non-linear Absorption and Scattering Processes in Solids

As indicated in Fig. 15.23a, the diffracted signal consists of a magnetic part, determined by the domains and a charge part due to the aperture and the homogeneous charge distribution in the film, which according to Fig. 8.12b only attenuates the intensity of the first order diffraction pattern. For the employed linear polarization, the magnetic and charge patterns are separable [81] as indicated in the figure, previously discussed in Sect. 8.7.4. In Fig. 15.23b, c we show the change of the diffraction pattern with increasing fluence, recorded by the CCD, of both the magnetic domains and the circular aperture [12]. Corresponding radial line scans of the q-dependent intensity distribution are shown in (d) and (e). The experimental intensity profiles are compared to simulations using the theory in Sect. 15.9.2 for the change of the Airy rings, whose decrease with intensity was previously shown in Fig. 15.20. Owing to the used beamstop, indicated in Fig. 15.23a, which prevented the saturation of the CCD detector, the expected intensity enhancement in the forward direction could not be observed in this experiment [12]. The incident pulses act on the atoms coherently for a time τ  20 fs which is much longer than the core hole decay time τ = 1.5 fs. For exact resonance excitation we can neglect the real part of the optical constants given in Sect. 15.10 (see Fig. 7.25a). p Furthermore, we can use the analytical solution of the Bloch equations for ρ

22 (τ ) p given by (15.75) to obtain βNL (τ ). The out-of-beam diffraction pattern in the presence of stimulation, normalized to the spontaneous diffraction pattern, is then given by 

stim Idiff spon Idiff

= q>0

1− e−2(β−βNL )kd ( β − βNL )2 , 1− e−2βkd ( β)2  

 

Fspon

(15.129)

CNL /Cspon

where CNL and Cspon are defined in (15.126). The normalized diffracted intensity given by (15.129) is plotted in Fig. 15.24 as a red curve superimposed on the experimental data points of [12]. We have also included in blue the simultaneously recorded decrease of the outer Airy rings, previously shown in Fig. 15.20c. The blue curve is given by Fspon alone in (15.129). The difference between the magnetic domain response (red) and the Airy ring charge response (blue) is solely due to the term CNL /Cspon in (15.129). There is good agreement between theory and experiment.

15.10.3 Broad Bandwidth Case In order to demonstrate the effect of temporal coherence in resonant magnetic diffraction, we have also included in Fig. 15.24 in green color the diffraction part of the study by Chen et al. [13], whose experimental geometry and principal results were previously shown in Fig. 1.30. In this case, the sample was not enclosed by a circular aperture and hence the diffraction pattern was entirely due to the magnetic domains

15.10 Polarization Dependent NL Diffraction

833

Diffraction rel. to spontaneous

Non-linear change of diffraction contrast 1.0 0.8 0.6 0.4

Co L3 resonance magn. speckle coherent pulse Airy charge rings coherent pulse magn. speckle SASE pulse

0.2 0.0 0.01

0.10

1

10

1000

100 2

Incident intensity I0 (mJ/cm /fs) Fig. 15.24 Change of the diffracted magnetic speckle (red) and Airy ring charge (blue) contrast as a function of incident intensity for a Co/Pd multilayer containing 20 nm Co in a circular aperture, resonantly excited with monochromatized 50 fs SASE pulses of linear polarization [12]. The plotted data points and error bars are from [12], representing the ratio of the total NL contrast normalized by the spontaneous contrast. The solid blue curve is taken from Fig. 15.20c and corresponds to the Fspon term alone in (15.129). The red curve is calculated with the same parameters according to the full expression (15.129). In both cases the coherence time was taken to be τcoh = 20 fs. The green data points represent averaged magnetic diffraction intensities recorded for similar Co/Pd samples with 25 fs SASE pulses [13]. The simultaneously recorded central transmission increase was previously shown in Fig. 15.15. The green curve is a simulation discussed in the text

in a film that was uniform from a charge distribution point of view. The intensitydependent transmission of the central beam was already discussed in Sect. 15.7.1, and the results were shown in Fig. 15.15. In contrast to the monochromatic pulses used for the data shown in red (magnetic diffraction) and blue (Airy ring or charge diffraction), the experimental green data bars in Fig. 15.24 correspond to 25 fs SASE pulses, whose internal structure in the energy domain was shown in Fig. 15.14b.9 In particular, the red and green curves and data points in Fig. 15.24 both represent the change in magnetic diffraction. The red curve represents the change in diffraction for an incident pulse of narrow monochromatized bandwidth (0.2 eV) tuned to the peak of the L3 resonance whose experimental width is about 1.6 eV as shown in Fig. 10.17b. Hence it optimizes the pure resonance response since all incident photons are tuned to the resonance. The green curve is shifted by a factor of about 30 to higher intensity relative to the red curve. This shift is larger than that expected from the ratio of the temporal coherence lengths alone, 20/1.8  11. In contrast to the narrow monochromatized bandwidth of 0.2 eV used for the red data, the green data were recorded with SASE pulses of large 4 eV energy width (see Fig. 15.14b). The additional shift is due to the fact that most of the incident 9

The experimental arrangement shown in Fig. 15.14a was actually part of a more complete arrangement shown previously in Fig. 1.30.

834

15 Non-linear Absorption and Scattering Processes in Solids

photons are not tuned to the resonance, so it takes a larger I0 to create a significant upper state population that creates the NL loss of magnetic diffraction. In particular, when the SASE bandwidth is larger than the width of the L3 resonance, the slightly off-resonance dichroic component ( δ)2 (see Figs. 7.25a and 8.30) also enters in addition to ( β)2 in the general magnetic diffraction term CNL given by (15.126) which more explicitly reads ' & CNL (ω) = [ β(ω)− βNL (ω)]2 + [ δ(ω)− δNL (ω)]2 k 2 d 2 .(15.130) The peak position of δ(ω) lies in the tail of the absorption resonance (i.e. of the β components), and therefore the sample is more transparent at that energy as pointed out in Sect. 8.11.5. This delays the increase of the NL component δNL with I0 , causing the spontaneous contribution δ(ω) to persist to higher intensities and delay the disappearance of magnetic diffraction. For the Co L3 case, the 4 eV energy width also covers both the position of the XAS peak at 778 eV and the position of the XES peak at 776 eV (see Fig. 13.15). The incident SASE pulse can therefore stimulate both REXS and RIXS decays. The green curve in Fig. 15.24 was calculated by accounting for the δ(ω) diffraction contribution, and taking into account that for Co metal the stimulated REXS channel is reduced by a factor of 2 because of an equally strong stimulated RIXS channel. This will be shown in the following section.

15.11 Stimulated Resonant Inelastic X-Ray Scattering In the following sections we will discuss the intensity-dependent evolution of the RIXS process, which combines the XAS “up” and XES “down” processes as previously discussed in Sect. 13.10.3.2. We will begin by outlining the concept of stimulated RIXS and review its experimental observation in Co metal by Higley et al. [7]. We then use a simple extension of the BR theory to describe the REXS and RIXS channels and their saturation. In the end, we will compare our theoretical formulation to the Co L3 data and their interpretation in the Kramers-Heisenberg-Dirac (KHD) formulation given in [7].

15.11.1 Stimulated L3 REXS and RIXS for Co Metal The REXS and RIXS processes associated with L3 excitation in Co metal are illustrated in a configuration picture (see Sect. 13.2.2) in Fig. 15.25a, and the dipole matrix elements for the up/down 2 p ↔ 3d transitions are shown in (b) in a one-electron picture.

15.11 Stimulated Resonant Inelastic X-Ray Scattering

835

(a) Configuration REXS/RIXS picture

3d

|c 3d band model

2p3/2

3d holes DOS

k

3d electrons

h

REXS RIXS 1

h

1

h

2

3d

|b

3d

2p3/2

|a

2p3/2

(b) One-electron picture: matrix elements Dipole transitions 2p c=1

valence states

nh empty orbitals ne filled orbitals

3d L=2

Co metal

nh= 2.5/2 ne= 7.5/2

nh+ne= 2L+1

L nh =1/6 3 2L+1

core states

ne L =1/6 3 (2L+1) (2c+1) one empty orbital

Fig. 15.25 a Configuration-based transitions for L-shell 2 p ↔ 3d transitions between core and valence states for transition metals. Empty valence states are shown as empty boxes and filled valence states as filled boxes. Here |a is the electronic ground state, |c the intermediate coreexcited state, and the final states are either the ground state |a (REXS, black) or an exited valence state |b of energy E above the ground state (RIXS, red). b Orbital degeneracies (without spin) of the electronic states involved in the transitions, and values of the angular transition matrix elements that take into account the degeneracies. For Co metal, there are 2n h = 2.5 empty 3d spin orbitals and 2n e = 7.5 filled 3d spin orbitals, as indicated [38]. This leads to the same up/down angular dipole matrix elements. The radial 2 p ↔ 3d dipole matrix elements are also the same

836

15 Non-linear Absorption and Scattering Processes in Solids

Figure 15.25a summarizes previous diagrams, shown for XAS in Fig. 10.14, XES in Fig. 12.9, and RIXS in Fig. 13.15, and require no detailed discussion. For the Co L3 case, the center of the XAS and REXS transitions is located at 778 eV, and the XES and RIXS transitions are centered at 776 eV as shown by the spectra in Fig. 13.15b. The key message of Fig. 15.25b is that for Co metal, all up/down 2 p ↔ 3d total transition matrix elements work out to be the same. The angular parts are the same because of the compensating degeneracies of the lower and upper states, as illustrated in the figure, and the radial matrix elements do not depend on the degeneracy or filling of the core and valence states involved and are therefore also the same (see Sect. 12.4.1). The L3 (2 p3/2 ) fraction of the total L-shell contribution is simply 2/3. The REXS and RIXS processes therefore are seen to have equal probabilities. The two channels are furthermore independent and do not interfere because the states |a and |b are orthogonal. These considerations are indeed born out by experimental results which we will discuss now.

15.11.2 Observation of Stimulated RIXS in a Solid The relevant part of the complete experimental arrangement used by Higley et al. [7] is that shown in Fig. 15.14. The transmitted photons were energy separated and detected by an in-line spectrometer with an energy resolution of about 0.8 eV. The photon energy of the about 4 eV wide incident SASE pulses was centered around the Co L3 absorption resonance of 778 eV to produce Co 2 p3/2 core holes. At low incident fluence, the transmitted intensity distribution recorded by the in-line spectrometer is the dispersed XAS spectrum. The intensity structure of the incident pulses was normalized out by the pulse splitting scheme in Fig. 15.14. A typical transmitted intensity measured in XAS is about 30% of the incident intensity. In contrast, the spontaneous REXS and RIXS intensities are not only much weaker, but they are also emitted into a 4π solid angle. An upper estimate of the spontaneous RIXS intensity seen by a spectromter with finite acceptance angle is obtained by the product of the fluorescence yield Yf times the solid angle of acceptance of the RIXS spectrometer, which is of order 10−5 of 4π steradian [82–86]. In our case, the Co LVV x-ray emission has a fluorescence yield of Yf = 8 × 10−3 so that the spontaneous RIXS signal is only of order 10−7 − 10−6 of the transmitted signal.10 This reveals the near impossibility of measuring the tiny spontaneous RIXS component of the total transmitted intensity. In practice, spontaneous RIXS is therefore measured which a spectrometer whose acceptance cone is at an angle with the direct beam. On the other hand, the transmission geometry is ideally suited to demonstrate the orders of magnitude enhancement of RIXS upon stimulation by the incident beam itself. The stimulated enhancement is similar to that of REXS, and the stimulated REXS and RIXS signals emitted in the forward direction can be 10

The L3 contribution is given by 2/3 of this value, which is the core state degeneracy ratio 2 p3/2 /(2 p3/2 + 2 p1/2 ) = 4/6.

15.11 Stimulated Resonant Inelastic X-Ray Scattering 0.3 0.25

(a) spont. RIXS

0.8

spont. transmission

0.6 0.4

Relative intensities

0.2 0

0.5 0.4 0.3

(b) spont. transmission non-linear transmission

reference pulse intensity

0.2

0.1 0.05 0 -0.05 0.25 0.2 0.15

reference x 0.5 non-linear change non-linear gain

non-linear loss

0.1 0.05 0 -0.05 0.25 0.2 0.15

0

0.1 0.05 0

770 772 774 776 778 780 782 784 786

-0.05

0.1

Photon energy (eV)

(d)

(c)

0.2 0.15

Co L3

spont. XAS

Non-linear minus spontaneous transmission

1

837

770 772 774 776 778 780 782 784

772 774 776 778 780 782 784 786

Photon energy (eV)

Fig. 15.26 a Comparison of the scaled spontaneous Co L3 RIXS spectrum (gray) for Co metal [87] and the Co/Pd XAS spectrum (black), both recorded at synchrotron light sources (low-fluence limit). The red spectrum is the transmission version of the XAS spectrum. b Example of data extraction and normalization. The dashed gray line is the reference spectrum of a 25 fs pulse of 9490 mJ/cm2 fluence transmitted through the SiN window, multiplied by 0.55 to account for the constant non-resonant absorption of the Co/Pd sample. The blue curve is the measured transmission spectrum through the Co/Pd sample at the stated high fluence. The red curve is the spontaneous (low fluence) transmission spectrum, obtained by multiplying the red spectrum in (a) by the dashed gray reference spectrum. Light blue shaded areas indicate non-linear gain and red areas non-linear loss. Panels (c) and (d) show as dashed lines the reference spectra transmitted through the SiN for 5 fs and 25 fs pulses for different pulse shapes and fluences. The associated transmission difference spectra are shown as solid black lines. They were obtained by subtraction of the spontaneous low-fluence spectra from the non-linear high-fluence spectra for the respective transmission curves. The shading of areas corresponds to the procedure (blue minus red curves) illustrated in (b). Each spectrum is an average of many shots as discussed in the text. The centers of three regions with non-linear response are denoted by dashed vertical lines and labeled α, β, and γ

directly compared by use of an in-line spectrometer capable of separating the photon energies (2 eV for Co) of the two signals. Experimental results of [7] are shown in Fig. 15.26. Panel (a) shows the spontaneous RIXS (gray) and XAS (black) spectra recorded with synchrotron radiation taken from Fig. 13.15b. The shown spectra are arbitrarily scaled to the same unit peak value. Also shown as a red curve is the corresponding spontaneous transmission spectrum with a 32% transmission at the L3 resonance. Panel (b) illustrates the extraction of the transmission difference spectra to obtain the NL relative to the spontaneous response. The dashed gray line is the reference transmission spectrum of a 25 fs pulse. The red curve is the calculated spontaneous (low fluence) transmission spectrum through the Co sample, obtained by multiplying the red curve in (a) by the reference pulse transmission spectrum. The blue curve

838

15 Non-linear Absorption and Scattering Processes in Solids

is the transmission spectrum measured for the indicated pulse length and fluence. The shaded areas highlight the NL changes in transmission, with light blue areas indicating NL transmission gain and red areas NL transmission loss. The top row in Figs. 15.26c, d demonstrates the quality of the normalization procedure. The shown data for both 5 fs and 25 fs pulses correspond to multiple shots that were binned according to the central photon energy of the x-ray pulses. For each case, the dashed gray curves are the reference pulse spectra, scaled by 0.5 to emphasize the difference spectra shown as solid black lines. The low-fluence spontaneous transmission curves in the top row are identical within noise. In the lower rows of Fig. 15.26c, d, a NL response is found in the three regions indicated by vertical dashed lines. The central and bottom rows, respectively, show high-fluence difference spectra, centered at different photon energies. The feature γ around the XAS peak position at 778 eV appears prominently as a transmission gain (blue) in all difference spectra. In contrast, the NL features α and β show different behavior when the incident fluence distribution is shifted. As shown in the bottom row, the α feature disappears and the β feature becomes stronger when the incident distribution shifts to higher energy. Figure 15.27 shows a simple modeling of the observed results by assuming three types of non-linear effects. Stimulated REXS and RIXS are accompanied by a contribution due to electron redistribution previously shown in Figs. 15.3a, b. In Fig. 15.27a we show the assumed spontaneous RIXS (gray), XAS (black), and transmission (red) spectra. The RIXS spectrum has been arbitrarily scaled to unit peak height. Also shown are the relative energy distributions and sizes of the three NL contributions, assumed to represent those at the highest incident fluences in Fig. 15.26. We assume that the stimulated RIXS (magenta) and REXS (blue) contributions have the shape of the spontaneous spectra in (a) and have the same size of 18% of the resonant XAS peak value and area. The electron redistribution (green) is modeled by the difference of two Fermi-Dirac distributions as illustrated in Fig. 15.1c. It has a peak value of 20% of the resonant XAS value and 8% of the integrated XAS area. Depending on the energy distribution of the incident pulse, the three NL contributions contribute with different shapes and intensities as shown in Figs. 15.27b, c, modeled to reflect the two 25 fs high-fluence cases on the right side of Fig. 15.26. Finally, we show in Fig. 15.27d the change of the spontaneous transmission spectrum in (a), shown again in red, by adding to it the three NL contributions in Fig. 15.27b. The total NL transmission spectrum (black) exhibits strong NL effects whose spectral distortions are indicated by arrows. The close quantitative agreement of our simulations with experimental results for corresponding incident pulse distributions is underscored by their direct comparison on the same vertical scales in Fig. 15.28. Of particular importance is that the stimulated RIXS spectrum, although, partly obscured by electron redistribution, appears on the same intensity scale as the change in elastic transmission. This arises from a

1.0 0.8 0.6 0.4 0.2 0.0 -0.2

0.20 0.15 0.10 0.05 0.00 -0.05 -0.10

(a)

1.2 1.0 0.8 0.6 0.4

839

spont. XAS

scaled spont. RIXS

spont. transmission stim. REXS stim. RIXS electron redist.

(b)

pulse intensity distribution x 0.5

sum

stim. REXS stim. RIXS electron redist.

0.20 (c) 0.15 0.10 0.05 0.00 -0.05

NL transmission change

NL transmission contributions

Rel. intensities

15.11 Stimulated Resonant Inelastic X-Ray Scattering

pulse intensity distribution x 0.5

sum

EF

(d) stimul. RIXS total NL transmission spont. transmission

electron redist.

stimul. REXS

772 774 776 778 780 782 784 Photon energy (eV) Fig. 15.27 a Model Co L3 spontaneous RIXS (gray), XAS (black) and transmission (red) spectra, and three assumed non-linear (NL) contributions, stimulated RIXS (magenta), electron redistribution (green), and stimulated REXS (blue). The sizes of the NL contributions are referenced to the unit value of the XAS peak, and their shapes are discussed in the text. b Assumed incident pulse reference distribution (dashed), which allows all three NL channels to contribute to the sum shown in black. c Shifted pulse reference distribution (dashed) which eliminates the stimulated RIXS contribution. The remaining two add up the sum shown in black. d Change of the spontaneous transmission spectrum (red) taken from (a), to the total NL transmission one (black) for a wide incident energy distribution. The black curve is the red curve plus the the sum of all three NL contributions. Colored arrows indicate NL transmission gain (up arrows) and loss (down arrows) caused by the respective NL channels

840

15 Non-linear Absorption and Scattering Processes in Solids

Transmission change (non-linear minus spont.) reference

0.2

(a)

(b)

Experiment

Simulation

0.15 0.1

scaled pulse intensity distribution

25 fs

0.05 0 -0.05

non-linear gain

0.2

non-linear loss

0.15 0.1 0.05

25 fs

0 -0.05 772 774 776 778 780 782 784

772 774 776 778 780 782 784

Photon energy (eV)

Fig. 15.28 a Experimental high-fluence non-linear effects for 25 fs pulses taken from Fig. 15.26d. b Simulated results with our model for similar incident intensity distributions, plotted on the same scale. The shown simulated curves are the black curves in Fig. 15.27b, c

stimulated amplification of the weak spontaneous RIXS intensity by a factor of order 106 (see below). The good quantitative agreement of theory and experiment shows that the stimulated RIXS and REXS are about the same for 25 fs pulses of fluence in the range 9 − 9.5 × 103 mJ/cm2 or an intensity around 350 mJ/cm2 /fs. At this value, the size of the stimulated RIXS intensity has a value of 18% of the spontaneous XAS intensity. In the following we compare these values with calculations for the stimulated RIXS rate.

15.11.3 The Stimulated REXS/RIXS Model The approximately 65% peak transmission at an intensity around 350 mJ/cm2 /fs, predicted by our model shown in Fig. 15.27d, agrees well with the value at the same intensity previously shown in Fig. 15.15. This is illustrated by the direct comparison in Fig. 15.29. Saturation occurs when at high incident intensity the population of the upper state, which is the intermediate state |c in Fig. 15.25a, reaches 1/2. When the incident photons have an energy distribution wider than E in Fig. 15.25a, they can stimulate both REXS and RIXS decays. In the forward direction, the “impulsive” stimulation by the incident beam itself can lead to equally strong REXS and RIXS intensities. The respective cross sections [see (15.43) and (15.46)] are then given by

15.11 Stimulated Resonant Inelastic X-Ray Scattering

(a)

1.0 0.8

NL transmission ~350 mJ/cm2/fs

0.6

Transmission change

Fig. 15.29 a Spontaneous and NL transmission spectra taken from Fig. 15.27d, with horizontal dashed red and blue lines indicating the change in peak transmission values. b Transmission curve replicating that in Fig. 15.15 in the same color

841

0.4

spont. transmission

772 774 776 778 780 782 784 Photon energy (eV) 1.0

(b)

0.8

BR analytical =0.7-3.0 fs

0.6 0.4 0.2 1

10

100

104

1000

Incident intensity I0 (mJ/cm /fs) 2

♦ tot NL σXAS = σXAS − σXAS NL σXAS NL σRIXS

= =

NL NL σRIXS + σREXS ♦ NL σREXS = σXAS ρ

22 (τ ).

(15.131) (15.132) (15.133)

The distinction between the REXS and RIXS photons whose energy only differs by E was not considered in the study by Chen et al. [13], so that the change in transmission illustrated in Fig. 15.15 was attributed to stimulated REXS alone. As revealed by a more detailed data analysis by Higley et al. [7], the stimulated REXS and RIXS channels contribute equally as shown in Fig. 15.27a with an additional contribution from electron rearrangement around the Fermi level that is included in Fig. 15.29a.

15.11.3.1

Photon Stimulation Versus Electron Reshuffling in RIXS

The BR theory shown in Fig. 15.29b properly accounts for the peak transmission shown in (a), although it does not explicitly include electron-electron reshuffling. Our BR formulation of stimulated REXS/RIXS scattering corresponds to the direct RIXS model (see Sect. 13.2.4) for the intermediate state. In this formalism the atomic

842

15 Non-linear Absorption and Scattering Processes in Solids

response is effectively described in a one-electron picture as the transition of a single active electron between atomic states. One may envision the electron reshuffling contribution to arise from an indirect RIXS process [88–91], where the decay is accompanied by an intra-valence band shake-up that reshuffles electrons around the Fermi level. The treatment of the indirect RIXS case is considerably more complicated, similar to the description of inelastic electron-electron scattering of Auger electrons. It is conceivable that the two pictures are linked in higher order QED. The lower intensity results of Sect. 15.3.1 together with our present higher intensity results suggest that resonant NL photon-solid interactions may first be dominated by valence electron redistribution (indirect RIXS), with the photon stimulation effect (direct RIXS) increasing with photon degeneracy parameter.

15.11.3.2

The Stimulated RIXS Gain

We can now quantitatively compare the stimulated RIXS gain in the forward direction measured by Higley et al. [7] with our theoretical model. This is done in Fig. 15.30, where we show as magenta (5 fs) and green (25 fs) filled circles the RIXS probabilities taken from [7] at the highest measured intensity of about 350 mJ/cm2 /fs. The plotted probabilities are defined by the ratio of the RIXS and the spontaneous XAS intensities. The spontaneously XAS intensity, which served as a reference, is shown on top as a black horizontal line of unit value. The lowest horizontal line in Fig. 15.30 indicates the L3 fraction of the fluorescence yield (FY) given by Yf (2 p3/2 ) = 23 Yf  5.3 × 10−3 , which corresponds to Stimulated RIXS rel. to spont. XAS probability rel. spont. XAS probability max. rel. RIXS probability

Stim. RIXS / XAS rate

1

-1

10

exp. exp. 5 fs 25 fs

BR analytical =0.7-3.0 fs

-2

10

KHD

spont. L3 RIXS rate into 4

pk=8.0eV

-3

10

1

10

100

10 3

Incident intensity I0 (mJ/cm2/fs) Fig. 15.30 Comparison of the I0 -dependent experimental and theoretical RIXS probabilities, NL /σ spon . Experimental data points around I = 350 mJ/cm2 /fs, taken from [7], are defined as σRIXS 0 XAS shown as magenta (5 fs pulses) and green (25 fs) data points. The red theory curve is calculated with (15.134), the dashed red curve with (15.136) and the blue curve with (15.138), with parameters discussed in the text

15.11 Stimulated Resonant Inelastic X-Ray Scattering

843

the spontaneous L3 RIXS signal emitted into 4π. In practice, only a tiny fraction, ∼10−5 of 4π, of this intensity falls into the detector acceptance cone. The main gain of the RIXS intensity seen by the detector is due to the fact that the stimulated RIXS intensity remains within the direction of the incident photons. It is therefore directly driven into the detector. The maximum stimulated RIXS intensity is only 1/2 of the spontaneous XAS value, as indicated by another horizontal line. Stimulated RIXS and REXS each contribute half of the total NL intensity. By comparison to the maximum value of 1/2, indicated by a horizontal line in Fig. 15.30, we see that at I0 = 350 mJ/cm2 /fs the calculated stimulated RIXS intensity shown as a solid red curve has nearly saturated. The solid red curve in Fig. 15.30 was calculated assuming the Co metal value ♦ = 6.25 Mb or transition matrix element x = 0.34 meV with the cross section σXAS given by, ♦ ♦ NL BR theory : σRIXS = σXAS ρ

22 (τ ) = σXAS

Geff ( R )2 /4 g(τ )

2 /4 + Geff ( R )2 /2

(15.134)

The excited population ρ

22 (τ ) is given by (15.72) with p = 0 or (15.75), with the time-dependent part expressed by     24 ωR 1−cos(ωR τ ) e−3 τ/(4) − 16(ωR )2 −9 2 sin(ωR τ ) e−3 τ/(4)   . g(τ ) = 1− ωR τ 16(ωR )2 + 9 2 (15.135) As for the curves in Figs. 15.15 and 15.29b, the g(τ ) expression was integrated over the range 0.7 ≤ τ ≤ 3.0 fs, resulting in the solid red curve in Fig. 15.30. The dashed red curve in Fig. 15.30 was calculated in the low intensity approximation R and therefore does not exhibit saturation. As shown for the atomic case by (14.44), one can obtain a time-dependent analytical solution at low intensity which contains the detuning energy E = ω − E0 . By use of R2 = n x (see (15.58)) and integrating (14.44) over time we obtain the low intensity expression of (15.134) in the same time notation τ as

x

2 /4 ♦ NL = σXAS Geff n

h(τ ), BR theory for R : σRIXS

(ω−E0 )2 + 2 /4  

[ρ22 (τ )]atom

(15.136) where we have identified by an underbracket the low intensity single atom population [ρ22 (τ )]atom , with the curly brackets in (14.44) integrated up to a time τ given by h(τ ) = 1+

       1− e− τ/  e− τ/(2) 2 E τ E τ E  e τ/(2) − cos −  2 + sin τ



 τ E + 2 /4

(15.137)

844

15 Non-linear Absorption and Scattering Processes in Solids

The factor Geff in (15.136), which for Co L3 excitation is Geff = 219 (see Table 15.2), is seen to simply multiply the actual population of the individual atoms in the film. For solids, both REXS and RIXS are enhanced in the forward direction by the coherent enhancement factor Geff  λ2 ρa deff /(4π ) in (15.40). The dashed red line in Fig. 15.30 was calculated with (15.136) for E = 0 with the same integration over τ as for the red curve. By comparison of the two curves we see that the validity range of the low intensity BR theory is limited to the overlap region I0 < 100 mJ/cm2 /fs) of the red and dashed red curves in Fig. 15.30. Finally we show as a blue line in Fig. 15.30 the result of the simple KHD direct RIXS formulation (13.83) used in [7] given by NL = npk KHD theory : σRIXS

x at λ3 at I0 Yf σXAS , σXAS =

c pk

(15.138)

where on the right side we have used n pk = λ3 I0 /(c pk ) (see (15.58)) and identified the fluorescence yield Yf with x / . In [7], the stimulated response was calculated with (15.138) by use of a Lorentzian atomic cross section to avoid the complications of solid-state effects, which enter through changes of the resonance width and intensity (see Fig. 15.10), exponential intensity decay through the film and a collective enhancement factor Geff . The blue at = 41 Mb correcurve is calculated with (15.138) and the parameters of [7], σXAS sponding to x = 2.2 meV, and a bandwidth of pk =8 eV, except we used the value Yf (2 p3/2 ) = 23 Yf  5.3 × 10−3 . Similar to the red dashed curve, the blue curve does not saturate and somewhat fortuitously predicts the magenta and green experimental values. It is interesting to compare the low intensity BR expression (15.136) for a solid 2

with the atomic KHD expression (15.138). By use of the relationship n pk = n 2π pk given by (14.80) which links the number of photons n of bandwidth in the atomic coherence volume with the number of photons n pk of bandwidth pk in mode volume, we obtain at resonance

x ♦ h(τ ) Geff σXAS

x 2π 2 at = n

σ .

pk XAS

NL [σRIXS ]BR = n

NL [σRIXS ]KHD

(15.139)

By comparison we obtain ♦ h(τ ) Geff σXAS  

time-dep.BR theory coll. atom (solid) response

=

2π 2 at σ pk XAS  

bandwidth-dep. KHD theory indiv. atom response

(15.140)

References

845

With the earlier given parameters and Geff = 219 (see Table 15.2) the left side of (15.140) yields the dashed red line in Fig. 15.30 and the right side the blue line. As has been discussed in Sect. 14.6.2 for the atomic REXS case, one may consider the stimulated RIXS cross sections in (15.139) as linking the time-dependent low intensity BR theory with the bandwidth-dependent KHD perturbation theory. We conclude this chapter with the following statement. When atoms in a solid are exposed to a laterally and temporally coherent beam tuned to an atomic resonance, the effective excited state population is larger by a factor of Geff than the true excited state population calculated for individual (independent) atoms. The enhancement factor Geff accounts for the collective response of the coherently illuminated atoms.

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15 Non-linear Absorption and Scattering Processes in Solids

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Chapter 16

Quantum Diffraction: Emergence of the Quantum Substructure of Light

16.1 Introduction In this chapter we discuss an entirely quantum mechanical description of diffraction1 . We will introduce the remarkably simple new paradigm that diffraction patterns, long associated with wave interference, are instead direct signatures of the quantum states of light. In first order QED (corresponding to conventional quantum mechanics) this link has remained hidden leading to the wave-photon ambiguity. The ambiguity is shown to disappear in second order QED, where only the photon-based description can account for the observed diffraction patterns. Historically, the description of diffraction has been based on the magical Huygens– Fresnel principle, discussed in Sect. 8.3.3. Its acceptance is mostly based on the fact that it gives the right result for conventional diffraction experiments. It therefore embodies a pragmatic rather than fundamental approach to describe physical processes. The principle postulates the mysterious emission of spherical waves from “points” even when there are no real atoms that can either emit light or scatter it! Another fascinating aspect of this description is that the detection of the “intensity”, calculated as the absolute value squared of the interfering fields, is not specified. This ignores the well-known fact that, in practice, the pattern is recorded as a mosaic of local “bright spots” as shown in Fig. 8.10, where a photon energy packet ω is taken from the total wave field. The simplest quantum formulation of diffraction is due to Feynman [1] who explained Young’s double-slit experiment by use of his space-time probability amplitude formulation of quantum mechanics (QM) [2]. His treatment of diffraction may be viewed more generally as a demonstration of the inherent wave-particle duality underlying various formulations of quantum mechanics [3], first expressed through de Broglie’s hypothesis that all matter has wave properties [4]. Relative to other formulation of QM, Feynman’s formulation (see Sect. 4.8) is particularly appealing since single photon probability amplitudes closely resemble classical wave fields [5, 1

It is based on a paper published on March 28, 2020 on the physics archive arXiv:2003.14217.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0_16

849

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16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

6]. Rather than resolving the wave-particle conflict, Feynman’s treatment effectively consolidated it. All formulations of diffraction within the confines of conventional QM are limited, however, by its well-known linearity [7]. In particular, QM may be viewed as a first order perturbation within the complete theory of light and matter, quantum electrodynamics (QED), which extends to infinite order. Of the three different formulations of QED by Tomonaga [8], Schwinger [9], and Feynman [10], it is again Feynman’s formulation that is most appealing and of practical utility, as pointed out by Dyson [11] in showing their equivalence in 1949. In particular, the concept of space-time probability amplitudes of single independent photons or electrons in QM may be extended to higher perturbative orders through the construction of probability amplitudes of an increasing number of particles. Remarkably, in first order the elementary building blocks of QED, photons and electrons, are described by the same probability amplitudes. This is the deeper reason why photons and electrons give the same conventional diffraction patterns [1, 12]. It is only in second order that the description of multi-particles states becomes different for Bosons (photons) and fermions (electrons), reflected for fermions by increasingly complex Feynman diagrams [11, 12]. The difference between the interference behavior of two Bosons and two fermions in second order QED is most beautifully demonstrated by a Hanbury Brown–Twiss type interference experiment carried out with two different charge-neutral isotopes of He atoms, Bosonic 4 He and fermionic 3 He [13]. Owing to their symmetric versus antisymmetric two-particle wave functions the Bosonic bunching peak is replaced by a fermionic antibunching tip. In Feynman’s formulation, the probability amplitudes of an increasing number of photons, corresponding to increasing orders in QED, are constructed by addition and multiplication of those of individual photons [5, 6]. The formulation for different cases is augmented by rules regarding the addition versus multiplication of single particle amplitudes [6, 12]. These rules become increasing complex for more than two photons. It is then advantageous to use the operator-based method based on the concepts of modern quantum optics [14, 15], introduced for the one-photon case in Sect. 4.7 and extended to the two-photon case in Sect. 5.5. This method pioneered by Glauber describes light in terms of orders of coherence O = 1, 2, 3 . . . ∞ [16] which are equivalent to the orders of perturbation in Feynman’s formulation of QED [6, 12]. In Glauber’s formulation, the construction of probability amplitudes for increasing number of photons is facilitated by the use of products of an increasing number of photon birth and destruction (detection) operators that can create and destroy quantum states containing increasing numbers of photons. This leads to the link of quantum states and diffraction patterns, the central theme of the present chapter. The existence of quantum states of light containing specific number of photons N was first experimentally verified by experiments in the late 1980s where individual photons (N = 1) [17] or photon pairs (N = 2) [18] were sent through a lossless beam splitter and their emergence from different output ports was examined by coincidence detection, as discussed in Sect. 5.2. In the process it became clear that Dirac’s famous

16.2 Generation of Different States of Light

851

statement that photons do not interfere with each other [19] holds only in first order QED. This was more directly revealed by diffraction experiments in the late 1990s where the conventional illumination of Young’s double slits was replaced by use of entangled photon pairs [20, 21], produced by parametric down conversion [22–24]. In this chapter, we utilize the concept of quantum diffraction to illustrate the new paradigm that within QED, diffraction images are direct signatures of different quantum states of light. This becomes apparent when the results of modern versions of Young’s double-slit experiment, performed by illumination with differently modified laser light and photon-based detection, are compared to the patterns predicted by the formulation of diffraction within QED. This direct link has remained hidden in the past because the treatment of diffraction by conventional quantum mechanics results in an accidental degeneracy of diffraction patterns for different quantum states. Diffraction has therefore continued to be explained by the ad hoc concepts of “coherent” and “incoherent” superposition of waves. Here we show that the degeneracy of patterns for different quantum states in first order is lifted upon extension of QED to second order. This evolution is shown to be particularly important since the wave-particle equivalence breaks down and the true photon-based nature of light emerges in the diffraction patterns. As a consequence, the wave theory of diffraction can in principle be abandoned altogether today, and the framework of statistical optics [25, 26] may be replaced by the more fundamental quantum formulation of light [15, 16]. In particular, the broad and difficult concept of “partial coherence” in wave optics can now be succinctly defined through the degrees of coherence of specific quantum states in different orders of QED and their diffraction patterns.

16.2 Generation of Different States of Light The advent of the laser has allowed the creation of different quantum states of light which are described by quantum optics [14, 15, 21, 27, 28]. In Fig. 16.1 we present different schemes that have been used to prepare double-slit-like sources. The reason for the shown order will become clear later when the cases will be linked to the evolution of their respective diffraction patterns from first to second order. Figure 16.1a shows the cases of “coherent” illumination of the double slits by a conventional source that has been made (first order) “coherent” by use of a monochromator and pinhole or by a laser which is higher order coherent. The first and second order diffraction patterns have been studied by Shimizu et al. [29] using either a monochromatized halogen lamp or a Ti:sapphire laser to illuminate the slits. On the right of the same figure we show the particular case where a strong incident near transform-limited pulse, whose width is approximately equal to the photon coherence time, is tuned to a well-defined atomic resonance in a thin film [30]. In the so-prepared source, absorption is compensated by stimulated emission [31], producing a second order coherent source [6].

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16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

Fig. 16.1 Experimental schemes of preparing different double-slit-like sources representing different quantum states, as discussed in the text. The double-slit dimensions are labeled as previously in Fig. 8.10

(a) “Coherent” and 2 nd order coherent sources

laterally “coherent” beam, wavelength

a

strong laterally “coherent” beam, wavelength

a

(c) Two single photon sources

(b) Entangled 2-photon sources

laser beam, wavelength

(d) phase-diffused sources

laser beam wavelength

resonant material

pulsed laser beam, wavelength

SPDC crystal

quantum dots

(e) thermal or pseudothermal sources phase modulator

chaotic beam or laser beam wavelength

moving rough glass plate

In Fig. 16.1b, a laser is used to illuminate a suitable thin crystal that through spontaneous parametric down conversion produces two spatially entangled photons, each with half the incident photon energy [32]. The “entangled biphoton” diffraction case has been extensively studied in the literature [5, 6, 21]. In Fig. 16.1c two single photons are simultaneously emitted from two quantum sources. This case can be implemented by use of a laser pulse that triggers two quantum dots or two trapped atoms or ions to simultaneously emit single photons [33–36]. In Fig. 16.1d the slits are illuminated by phase-diffused light, implemented by Liu et al. [5] by splitting a coherent laser beam and modulating the phase of one of the beams with a vibrating mirror. The intensity falling onto the two slits is kept constant so that the photons with random phases still obey Poisson counting statistics. Finally, as shown in Fig. 16.1e, the slits may be illuminated by chaotic light produced by a thermal source [37]. In practice, it is convenient to use higher intensity “pseudo-thermal” light generated by shining a laser on a rotating ground glass plate [38]. Such light exhibits both phase and intensity fluctuations with Bose–Einstein counting statistics [39]. It has been employed for double-slit diffraction by several groups [40–42].

16.3 The Formulation of Quantum Diffraction

853

The cases shown in Fig. 16.1 represent experimental schemes that generate specific quantum states of light whose characteristic first and second order diffraction patterns have been reported in the cited literature. In the following we shall illustrate how the observed diffraction patterns follow from the photon-based formulation of diffraction within QED, which directly links quantum states with their characteristic encoded diffraction signatures. The presented formulation of quantum diffraction is general up to second order in QED and covers the infinite number of quantum states of light within QED. These quantum states give rise to the complex light behavior typically described by the broad and difficult concept of “partial coherence” in statistical optics [25, 26]. The quantum formulation allows the specification of this concept by directly defining coherence and diffraction through that of individual quantum states. These states span the entire range between the limiting cases of coherent states represented by a Poisson distribution and chaotic states associated with a Bose–Einstein distribution.

16.3 The Formulation of Quantum Diffraction The experimental geometry for Young’s double-slit experiment is illustrated in Fig. 16.2a with identification of the relevant coordinates. The photons emerging from the two source points A and B, which may be expanded into slits, are detected at a single point or two points in a distant detector plane. The diffraction pattern is determined by the spatial variations in the number of detected quasi-monochromatic photons. The assumption of monochromaticity allows the reduction of the general space-time description of photons in QED to the spatial domain only, since the width of the photon energy distribution defines a time interval of photon arrival, given by the coherence time τ of the photons. This is taken into account in the design of diffraction experiments, as illustrated in Fig. 16.2b, c. The first order diffraction pattern is determined by the arrival probability of photons within the time increment τ at a given detection point. In first order, the classical and quantum patterns are the same and for the double slit are given by (8.20) which with q = kρ/z 0 reads I (ρ) = 4I0

    a2 ka k sinc2 ρ cos2 ρ , λz 0 2z 0 2z 0

(16.1)

where z 0 is the distance between the source and detector planes and ρ is the lateral distance from the center of the detector plane. The second order pattern is determined by the coincident arrival probability of two photons within τ at a single or two detection points. The first and second order diffraction cases previously discussed in Sects. 4.7 and 5.5 correspond to the lowest orders, m = 1, 2, of perturbation in QED. For our treatment below we write the diffraction probability in the form (5.69) or

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16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

(a) General geometry and coordinates detector plane

source plane

kA

A

rA B

x1

rB

1

x2

z0 2

kB

(b) First order double-slit geometry and detection pinhole detector or mirror 1

d

detector

|

1

-

2|

d pinhole or mirror

(c) Second order double-slit geometry and detection 2-photon detector 1-photon detector

1-photon detector

Fig. 16.2 a Assumed geometry and coordinates of photon propagation from points r A and r B in the source plane to distant points with coordinates ρ i or xi (i = 1, 2) in the detector plane. b In first order, photons emitted from either slit are detected at a single point using the scenarios shown in blue and red. This corresponds to the classical diffraction case discussed in Sect. 8.5.1. A photon detector is either scanned as a function of its separation ρ from the optical axis or a detector on the optical axis detects the probability that photons have taken paths through points ρ 2 = −ρ 1 , defined by scannable mirrors or pinholes. The detection probability is given by (5.69) with m = 1. c In second order, one measures the coincident arrival of two photons at a single or two points by use of the blue and red detection schemes. The detection probability is given by (5.69) with m = 2



 s |P(m) (ρ 1 , ρ 2 )|s = Pm G (m) (ρ 1 , ρ 2 ).

(16.2)

Here P(m) (ρ 1 , ρ 2 ) is the detection probability operator, given by the conventional quantum optical correlation function of order m and the triangular brackets denote the quantum mechanical expectation value, evaluated for a given quantum state |s  created at the source. As indicated on the right of (16.2), the diffraction pattern can be conveniently written in terms of a position-dependent shape function G (m) (ρ 1 , ρ 2 ) and an overall scaling factor Pm that is determined by conservation of emitted and detected photons. Before we evaluate the diffraction patterns for the double-slit case

16.3 The Formulation of Quantum Diffraction

855

we briefly review our previous first order results derived in Sect. 4.7 and second order ones from Sect. 5.5.

16.3.1 First Order Diffraction Formulation As derived in Sect. 4.7 the first order probability operator P(1) (ρ 1 , ρ 2 ) in (16.2) may be expressed either in the coordinates of the detection points ρ i or xi defined in Fig. 16.2a. For the case where the double slits are just two points, the detection probability is obtained as the expectation value of (4.124), or       1     (1) s P (ρ 1 , ρ 2 ) s = s X + Ys 4

(16.3)

where |s  is a given quantum state created in the source. The operators X and Y are given by (4.125) and (4.126). For the double-slit geometry in Fig. 16.2a the slit centers are separated by  and each has a finite width a. The detection probability is calculated according to (4.139) by integration over the slits according to 

 P(1) (ρ 1 , ρ 2 ) =

1 4a 2

k i (r ·ρ −r ·ρ ) ak† U akV e z0 U 1 V 2 drU drV .



 U,V slits

(16.4)

For a given quantum state, only certain matrix elements ak† U akV  will have nonzero values and they will determine the integration and hence the diffraction pattern (see Tables 16.2 and 16.3).

16.3.2 Second Order Diffraction Formulation For two source points, the second order detection probability is given by (5.14) or      (2)     1    s A + B + C + Ds s P (x1 , x2 ) s = 16



(16.5)

with the four operators A, B, C, D given by (5.15)–(5.18) and |s  being the state created in the source. For finite-sized slits our previous expression (5.42) takes the form

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16 Quantum Diffraction: Emergence of the Quantum Substructure of Light



 P(2) (ρ 1 , ρ 2 ) =

1 16a 4



 U,V,X,Y slits

k i (r −r )·ρ i k (r −r )·ρ ak† U ak† V akX akY e z0 U X 1 e z0 V Y 2

drU drV drX drY .

(16.6)

For a given quantum state, only certain matrix elements ak† U ak† V akX akY  will have non-zero values and they will determine the integration and hence the diffraction pattern (see Tables 16.4 and 16.5).

16.3.3 Order-Dependent Degree of Coherence In quantum optics, coherence is characterized by a degree of spatial coherence g (m) (x1 , x2 ) which depends on the order m of perturbation in QED. The previously defined degrees of first (m = 1) order (4.83) and second (m = 2) order (5.4) spatial coherence may be written in the form 

g

(m)

 P(m) (x1 , x2 ) (x1 , x2 ) =   m/2 . P(1) (x1 , x1 ) P(1) (x2 , x2 )

(16.7)

For the coherent cases, the numerators in g (m) (x1 , x2 ) factor into the denominators, so that the diffraction fine structure contained in both the numerators and denominators is normalized out, yielding a constant. It is then convenient to plot the normalized diffraction pattern G (m) (x1 , x2 ) defined through (16.2) which preserves the characteristic diffraction structure. We shall utilize both complementary formulations in the present chapter.

16.4 The Quantum States of Light The first and second order diffraction patterns, defined by (5.69) with m = 1, 2, are determined by quantum states involving two wavevector modes kA and kB as defined in Fig. 16.2a. In the following we will switch to the shorter and more convenient notation k = kA and k = kB . The two-mode quantum states produced in the cases shown in Fig. 16.1 involve different numbers of photons. In general, we distinguish collective states which contain an average number of photons per mode from states that contain a specific number of photons per mode. We first discuss the two-mode multi-photon collective states associated with Fig. 16.1a, d and e and their decomposition into probability distributions of substates. Their specific two-photon substates are then linked to the central two cases in Fig. 16.1b, c, which involve only two photons.

16.4 The Quantum States of Light

857

16.4.1 Two-Mode Collective Quantum States The two-mode collective coherent state produced in Fig. 16.1a and the phase-diffused coherent state in Fig. 16.1d are constructed from two single mode coherent states of the form (3.78) or ∞  αkm (16.8) |αk = √ |mk . |αk |2 /2 m! m=0 e Here αk is a complex number and the coherent state contains an average number of photons nk = |αk |2 in the mode k, distributed in a Poisson distribution around the average value |αk |2 . In general, the two modes may contain different numbers of photons and have different phases. In the following we shall assume that both modes contain the same average number of photons per mode, i.e. |α|2 = n = nk = nk  . We then obtain with αk = |α|eiφk and αk  = |α|eiφk  the following general expression for a two-mode collective “coherent” state |αk |αk  =

∞ ∞ 1   |α|n+m ei(nφk +mφk  ) |nk |mk  . √ e|α|2 n=0 m=0 n! m!

(16.9)

The two-mode chaotic state associated with Fig. 16.1e is constructed from two single mode chaotic states |βk given by (3.88) or |βk =

∞  m=0



nm |mk . (1 + n)1+m

(16.10)

It contains an average number of n photons per mode in the form of a Bose– Einstein distribution. The two-mode chaotic state containing an average number n of photons in each mode is given by |βk |βk  =

∞ ∞   =0 m=0

 e

i(φ +φm )

n nm |k |mk  . (1+n)+m+2

(16.11)

The states (16.9) and (16.11) are collective two-mode quantum states which contain the same average number of photons per mode, n = |α|2 . We now show that they may be written as a linear combination of two-mode substates that contain specific numbers of photons N = 0, 1, 2, 3 . . . ∞ with probability distributions around the mean value 2n = N , denoting the total number of photons in the two modes.

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16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

16.4.2 The Collective Coherent State and Its Substates The two-mode coherent state describes the case shown in Fig. 16.1a where the two slits are illuminated by the same average number of photons in both modes and the two modes have the same phases. We then have αk = αk  = |α|eiφ and the general expression (16.9) can be written in the form |coh k,k  =

∞ 1 

e|α|2 N =0

|α| N eiN φ

N  m=0

√

1 |mk |N −mk  . m! (N −m)!

(16.12)

The state is composed of binomial substates which contain specific numbers N of photons that are distributed according to a Poisson probability distribution around the average value 2n. We can write (see Sect. 5.6.1) |coh k,k  =

∞  2 N /2 α N |φcohN k,k  . √ ! e|α|2 N =0  N 

(16.13)

cαN

α 2 The complex coefficients cαN fulfill the normalization ∞ N =0 |c N | = 1 and weigh  the contributions of the binomial substates |φcohN k,k which contain N -photons and are given by |φcohN k,k  =

N 1 

2 N /2



m=0

N! |mk |N −mk  . m! (N −m)!

(16.14)

The states fulfill the normalization coh |coh  = φcohN |φcohN  = 1.

16.4.3 The Collective Phase-Diffused Coherent State and Its Substates The phase-diffused laser light encountered for the case in Fig. 16.1d corresponds to random phases between the two modes in (16.9). Since only the relative phase between the two modes is important, we may set φk = 0 and denote the relative phase shift as ϕ = φk  to obtain |dif k,k  =

∞ 1 

e|α|2

N =0

|α| N

N 

ei(N −m)ϕ |mk |N −mk  . √ m! (N − m)! m=0

(16.15)

16.4 The Quantum States of Light

859

This can be written in terms of substates containing a specific number of N photons with probabilities in form of a Poisson distribution around the average value 2n according to |dif k,k  =

∞  2 N /2 α N |φdifN k,k  , √ ! e|α|2 N =0  N 

(16.16)

cαN

where |φdifN k,k  =

N 1 

2 N /2

 ei(N −m)ϕ

m=0

N! |mk |N −mk  . m! (N −m)!

(16.17)

The states fulfill the normalization dif |dif  = φdifN |φdifN  = 1.

16.4.4 The Collective Chaotic State and Its Substates The two-mode collective chaotic state and its substates, describing the case in Fig. 16.1e, are given by the general form (16.11) which may be rewritten as |cha k,k  =

∞ 



N =0

N  n N ei(φm +φ N −m ) |mk |N −mk  . (1+n) N +2 m=0

(16.18)

The phase factors account for the relative phase difference between number states |mk and |N − mk  in the two modes. It also contains substates with a specific number of N -photons. Their probability is distributed in the form of a Bose–Einstein distribution around the average value 2n. The state (16.18) may be written as |cha k,k 

 ∞  (N + 1)n N = |φchaN k,k  , (1+n) N +2 N =0   

(16.19)

β

cN

where

∞

N =0

β

|c N |2 = 1 and the substates are |φchaN 

k,k 

=√

1

N 

N + 1 m=0

eiφm,N −m |mk |N −mk  .

The states fulfill the normalization cha |cha  = φchaN |φchaN  = 1.

(16.20)

860

16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

16.4.5 Plots of the Substate Distributions The substructure of the two-mode collective coherent, phase-diffused coherent, and chaotic states is illustrated in Fig. 16.3 for the cases of different orders of coherence O, defined by an average number of photons per mode O = n = 1, 2, 4, 9. The cases where 2n = N are shown in enhanced colors. They represent the total number of photons in the two modes. of all distributions sum to ∞ The βprobabilities α 2 2 |c | = |c | = 1. unity according to ∞ N =0 N N =0 N We distinguish the total average number of photons contained in the two-mode collective states 2n from the total specific number of photons N contained in the substates.

2

(a) “Coherent” states: |c N | = Probability P = |c N |

2

0.25

N

2 2 N! e

n

n

n n n n

N=2 n

0.20 0.15

N

=1 =2 =4 =9

0.10 0.05 0.00

0 5 10 15 20 25 30 Number of photons in both modes, N 2

2

(b) Chaotic states: |c N | = Probability P = |c N |

Fig. 16.3 Probability distributions of the substates of a the two-mode collective coherent state (16.13) and phase-diffused coherent state (16.16) and b the collective chaotic state (16.19). For the shown cases, the collective states contain different average number of photons per mode n = 1, 2, 4, 9, while their substates contain different total numbers of N -photons in both modes. The distributions peak approximately at N = 2n, indicated by enhanced colors, corresponding to the total number of photons in both modes

0.25 0.20 0.15 0.10

N=2 n

N +1 (1+ n )

N+2

n n n n n

N

=1 =2 =4 =9

0.05 0.00

0 5 10 15 20 25 30 Number of photons in both modes, N

16.4 The Quantum States of Light

861

16.4.6 Other Fundamental Quantum States 16.4.6.1

N-Photon Entangled or NOON State

A particularly important state in quantum information science is the N -photon entangled state [43–46] given by |φentN 

k,k 

  1 iφ   = √ |N k |0k + e |0k |N k 2

(16.21)

which we have written in a form that reflects why it is also called a NOON state. It corresponds to N -photons being emitted into the same mode and none into the other. The specific state N = 2 given by   1 |φent2 k,k  = √ |2k |0k  + eiϕ |0k |2k  2

(16.22)

describes the case in Fig. 16.2b.

16.4.6.2

N-Photon Number State

In the complementary case, an equal number of N /2 photons are emitted into separate modes, described by the two-mode number state |φnumN k,k 

    N N  =  . 2 k  2 k

(16.23)

Owing to the indivisibility of photons the state exists only for N ≥ 2. The specific state with N = 2 |φnum2 k,k  = |1k |1k 

(16.24)

describes the case in Fig. 16.2c.

16.4.7 Summary of Key Multi-photon Quantum States The normalized two-mode multi-photon states representing the cases with corresponding names in Fig. 16.1 are summarized in Table 16.1.

862

16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

Table 16.1 Quantum states of light associated with Fig. 16.1 a Coherent states and substates: ∞ 2 N /2 α N α |coh k,k  = ∞ N =0 c N |φcohN k,k  = N =0 √ N ! e|α|2 |φcohN k,k   N N!  |φcohN k,k  = 2 N1/2 m=0 m! (N−m)! |mk |N −mk Phase-diffused coherent states and substates: ∞ 2 N /2 α N α |dif k,k  = ∞ N =0 c N |φdifN k,k  = N =0 √ N ! e|α|2 |φdifN k,k   N 1 N! |φdifN k,k  = 2 N /2 m=0 ei(N −m)ϕ m! (N−m)! |mk |N −mk  Chaotic states and substates:  ∞ β (N +1)n N = c |φ  |cha k,k  = ∞ chaN k,k N =0 N N =0 (1+n) N +2 |φchaN k,k  N |φchaN k,k  = √ N1+1 m=0 eiφm,N −m |mk |N −mk  N -photon entangled (NOON) state:   |φentN k,k  = √1 |N k |0k  + eiϕ |0k |N k  2

N -photon number state (N > 1): |φnumN k,k  = |N /2k |N /2k  collective states |i k,k  have n photons in each mode, and for the coherent states we have |α|2 = n. The substates |φ j k,k  have N -photons in both modes. All states have unit normalization α 2 ∞ |cβ |2 = 1 and ∞ N =0 |c N | = N =0 N ∞ N β 2 N α 2 We also have the important sum rules in first order: ∞ N =0 2 |c N | = n and 2 |c N | = ∞ N (N −1) α 2 ∞ N (N −1)N =0 β 2 2. in second order: N =0 |c | = |c | = n N =0 N N 4 6 a The

16.5 First Order Double-Slit Diffraction Patterns   The key part of the calculation of the first order patterns P(1) (ρ 1 , ρ 2 ) given by (16.4) is the evaluation of the matrix elements or expectation values of the operators X and Y given by (4.125) and (4.126) with the quantum states in Table 16.1. The lengthy calculations of the matrix elements involve application of the well-known quantum mechanical rules for the action of creation and annihilation operators on the number substates [15, 19] and utilize sum rules like those listed at the bottom of Table 16.1. The matrix elements are derived in Appendix A.9 and for convenience are listed in Table 16.2 for the collective states containing an average of n photons per mode and in Table 16.3 for the states containing a total of N -photons in both modes.

16.5.1 Calculation of the First Order Patterns For the double-slit geometry in Fig. 16.2a, we have rA  rB  ρ 1  ρ 2 and the two slits of width a have a center-to-center separation of  = |rA − rB | = 2r . The integration in (16.4) over the slits consists of four integrals over source points rU and r V that

16.5 First Order Double-Slit Diffraction Patterns

863

Table 16.2 Matrix elements φx |O(1) |φx  = O(1) x of the four first order coherence operators O(1) with the collective quantum states of Table 16.1 containing an average number of n photons per mode or a total of 2n photons Operator O(1) O(1) coh O(1) dif O(1) cha ak† AakA

n

n

n

n

n

n

ak† AakB

n

0

0

n

0

0

ak† BakB

ak† BakA

Table 16.3 Matrix elements of the first order coherence operators O(1) with quantum states containing a total of N -photons in two modes Operator O(1) O(1) cohN O(1) entN O(1) numN O(1) difN O(1) chaN ak† AakA ak† BakB ak† AakB ak† BakA

N 2 N 2 N 2 N 2

N 2 N 2

N 2 N 2

N 2 N 2

N 2 N 2

0

0

0

0

0

0

0

0

may lie either in the same slit or in different slits. This accounts for correlations between photons born within the same or in different slits. To demonstrate how the patterns for two source points in changed upon integration over slits of finite size, we derive the results for the collective coherent state (16.12) and the collective chaotic state (16.18).

16.5.2 Coherent State For the collective coherent state, |coh , all matrix elements ak† X akY  for any two points X and Y are the same as seen from Table 16.2 and 16.3. They can hence be pulled out of the integral, and we obtain

864

16 Quantum Diffraction: Emergence of the Quantum Substructure of Light



(1)



P (ρ 1 , ρ 2 ) coh =

n 4a 2

 /2+a/2  ikr ρ e z0 U 1 drU /2−a/2

/2+a/2 

e

−i zk rV ρ2 0

drV

/2−a/2







upper slit −/2+a/2 

+

e

i zk rU ρ1 0

−/2+a/2 

drU

−/2−a/2

e

−i zk rV ρ2 0

drV

−/2−a/2







lower slit /2+a/2 

+

e

/2−a/2



i zk rU ρ1 0

−/2+a/2 

drU

e

−/2−a/2





upper slit −/2+a/2 

+

e

−/2−a/2



drV



/2+a/2 

drU

e

/2−a/2



0



lower slit

i zk rU ρ1 0

−i zk rV ρ2



lower slit

−i zk rV ρ2 0

 drV .





upper slit

(16.25) The integrals over the lower and upper slits are evaluated by use of

lower slit :

1 a

−/2+a/2 

e

±i zk r ρ 0

dr = e

k ∓i 2z ρ 0

−/2−a/2



ka sinc ρ 2z 0

 (16.26)

and upper slit :

1 a

/2+a/2 

e

±i zk r ρ 0

/2−a/2

dr = e

k ±i 2z ρ 0



 ka sinc ρ . 2z 0

(16.27)

The sinc functions factor and the exponential terms combine to cos-type modulations according to 

         k ka k ka ρ1 cos ρ2 sinc ρ1 sinc ρ2 .(16.28) P(1) (ρ 1 , ρ 2 ) coh= n cos 2z 0 2z 0 2z 0 2z 0      point like slits

finite slit width

The integrations over the finite slit widths are seen to give an additional sinc-type function due to the finite slit widths a that represents an envelope function of the finer cos-type modulation of point-like slits. For n = 1/2 per mode or a single photon in the two modes and point-like sources we obtain our previous result shown in Fig. 4.17b.

16.5 First Order Double-Slit Diffraction Patterns

16.5.2.1

865

Chaotic Case

For the collective chaotic state, only the matrix elements cha |ak† I akI |cha  = n with I = A, B listed in Table 16.2 are non-zero . There is no interference of photon probability amplitudes from different slits. In addition, the integral over each slit involves a correlation of the form δ(rU − rV ) between different points, since according to Table 16.2 only probability amplitudes from the same points within the slits can interfere. In this case (16.4) reduces to



(1)



P (ρ 1 , ρ 2 ) cha =

n 4a



−/2+a/2 

e −/2−a/2

i zk r |ρ 1 −ρ 2 | 0

dr







lower slit /2+a/2 

+

e

/2−a/2



i zk r|ρ 1 −ρ 2 | 0



 dr .

(16.29)



upper slit

The cross-slit interference terms are absent, and by use of (16.26) and (16.27) we obtain       (1) n ka k |ρ 1 − ρ 2 | sinc |ρ 1 − ρ 2 | . (16.30) cos P (ρ 1 , ρ 2 ) cha= 2 2z 0 2z 0      point like slits

finite slit width

For n = 1/2 per mode or a single photon in the two modes and a = 0 we obtain the pattern previously shown in Fig. 4.17c. When we write the above patterns as P(1) (ρ 1 , ρ 2 ) = P1 G (1) (ρ 1 , ρ 2 ) according to (5.69), we find that in both (16.28) and (16.30) the prefactor P1 scales with the average number n of photons/mode emitted by the source, but the shape of the patterns G (1) (ρ 1 , ρ 2 ) is independent of n. We can summarize as follows. The diffraction patterns for two finite-sized slits are those of two point-sized slits embedded in sinc-type envelope functions. The patterns scale with the average number of photons n emitted by the slits, but their shape is the same for a given quantum state, independent how many photons it contains. The patterns for the other quantum states in Table 16.1 are derived in Appendix A.9.

866

16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

16.5.3 Plots of the First Order Patterns The calculated shapes G (1) (ρ 1 , ρ 2 ) of all first order patterns for the quantum states in Table 16.1 (see Appendix A.9) are plotted in Fig. 16.4. For convenience, both the shapes and scaling factors P1 for the states are listed on the right. Remarkably, the different quantum states result in only two kinds of characteristic patterns, revealing a degeneracy of the patterns for several of the states. The patterns are identical to those derived by wave or statistic optics for the limiting cases of “coherent” and “incoherent” light [6, 26]. The coherent states and their substates produce the same characteristic pattern, shown as dashed thick blue and red curves in Fig. 16.4a for a = /4. The patterns are the same for the blue and red detection geometries in Fig. 16.2b. This is due to the   fact that the detection probability P(1) (ρ 1 , ρ 2 ) factors in the coordinates ρ 1 and ρ 2 [16]. For convenience we also show the two limits of point-like slits of width a  as a gray curve, and a single slit of width a, represented by the thin red-blue sinc2 envelope function.

(N >1):

Fig. 16.4 First order double-slit diffraction patterns for the quantum states of light in Table 16.1, with the blue and red colors representing the two detection schemes in Fig. 16.2b. The shape of the patterns G (1) (ρ 1 , ρ 2 ) and scaling factors P1 are given on the right. a Patterns G (1) (ρ 1 , ρ 2 ) for the coherent state and its substates representing the cases in Fig. 16.1a. The gray curve assumes point-like slits of width a , the thick red-blue pattern is that for a = /4, and the thin red-blue line represents the sinc2 envelope function which remains if the two slits are joined into a single slit of width a. b Patterns for all other quantum states and cases in Fig. 16.1b–e

16.5 First Order Double-Slit Diffraction Patterns

867

All other states in Table 16.1 form the patterns shown in Fig. 16.4b. The detection probability does not factor in the coordinates ρ 1 and ρ 2 , and the source is “incoherent” in first order. This results in constant photon counts for the blue detection geometry in Fig. 16.2b with a diffraction structure observed only for the red detection geometry. In all cases, the shapes of the patterns G (1) (ρ 1 , ρ 2 ) in Fig. 16.4 are the same for the three collective states containing an average of 2n photons and their respective substates containing N -photons. The shapes are even independent of the number of photons, N , contained in the substates. Emission of more than single photons, N > 1, only increases the scaling factor P1 , i.e. the overall detection probability. This explains the experimental fact that the first order pattern for the case shown in Fig. 16.1a is independent of whether the slits are illuminated by a thermal source that has been made first order coherent by energy (monochromator) and spatial (aperture) filtering or by a laser, although the light is coherent to different orders in QED for the two cases. This means that the nature of the collective states remains preserved in the coefficients and phases of their respective substates, independent of the number of photons, N , they contain. The patterns in Fig. 16.4 may therefore be recorded with one-photon detectors that pick out the N = 1 state or charge integrating detectors which detect arbitrary numbers of arriving photons. Note that even the collective states with n = 1 contain different number of photons N = 1, 2, 3 . . . as shown in Fig. 16.3. By use of charge or photon number integrating detectors the diffraction patterns therefore accumulate through the arrival of photons whose number statistically varies between N = 1 and about N = 6. This shows that the first order patterns do not depend on the photon degeneracy parameter or number of emitted photons per mode, n or N /2. This agrees with Dirac’s statement that photons do not interfere with each other [19]. The subtlety of this statement, however, is revealed by the fact that photons in the same mode are in principle indistinguishable as emphasized by Ou et al. [27, 28], and one would therefore expect them to interfere. The dilemma is resolved by the fact that a meaningful diffraction pattern always corresponds to the accumulation of a large number of detection events. Any interference structure that may be present when only few photons arrive in coincidence is increasingly averaged out upon appearance of a statistically meaningful pattern [47, 48]. In x-ray science, the independence of the Bragg diffraction pattern on the degeneracy parameter has the important consequence that all diffraction patterns observed for the last 100 years with weak sources such as Röntgen tubes remain the same when recorded in a single shot with an x-ray free electron laser, despite an increase of the degeneracy parameter by about 25 orders of magnitude [6]. This means that it does not matter whether the pattern assembles one-photon-at-a-time or in a “single shot”. With increasing degeneracy parameter the statistics of the pattern is simply improved. The pattern may be recorded much faster, even in a single few-femtosecond shot [49]. Hence no new x-ray diffraction theory is needed to describe first order patterns.

868

16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

16.5.4 Degree of First Order Coherence   The first order diffraction probabilities P(1) (ρ 1 , ρ 2 ) for the cases in Fig. 16.4 are readily converted by means of (16.7) into the corresponding degrees of first order coherence. For the double-slit geometry we have 

 P(1) (ρ, −ρ)  . g (ρ, −ρ) =  (1) P (ρ, ρ) (1)

(16.31)

For the coherent case in Fig. 16.4a we obtain with (A.51) the expected result g (1) (ρ, −ρ) = 1.

(16.32)

The “incoherent” cases in Fig. 16.4b all yield the same expression    k ka ρ sinc ρ . g (ρ, −ρ) = cos z0 z0 (1)



(16.33)

Hence we have g (1) (ρ, −ρ) = G (1) (ρ, −ρ) for these cases, represented by the red pattern Fig. 16.4b.

16.5.5 Reduction of First-Order Quantum to Wave Formalism The first order quantum formulation of diffraction gives the same patterns as the classical or statistical optics descriptions. In the quantum formulation, however, different diffraction patterns may directly be linked to different quantum states created in the source. The conventional wave formalism of diffraction emerges from the quantum descriptions by approximating the average over all photon emission directions by the concept of a spherical wave. One then makes the ad hoc assumption that all waves are created with the same birth phase. The distinction between coherence and incoherence arises from geometry alone. Spherical waves emitted from different source points are said to be coherent at a distant observation point, if one may approximate them by plane waves within a finite solid angle “observation cone” extending backward from the observation point to the source points. The picture may also be turned around by defining a coherent emission cone rather than a coherent observation cone. This simple geometrical concept means that within the solid angle coherence cone the curvature of the spherical wave is negligible on the scale of the wavelength. Classically one simply calculates the “intensity” at different points in the observation plane as the absolute value squared of interfering wave fields. The measurement process of the intensity remains unspecified.

16.6 Second Order Double-Slit Diffraction Patterns

869

One might have expected that an incoherent or chaotic source, reflected by the chaotic state in quantum optics, does not give rise to a diffraction pattern at all. The reason for its existence, revealed by the red pattern in Fig. 16.4b, lies in the fact that for the spatial phenomenon of diffraction one assumes space-time separability and temporal coherence, i.e. that the photons have the same well-defined energy or wavelength λ. If the source was chaotic in both space and time, there would indeed be no position-dependent diffraction structure. In wave or statistical optics, the pattern in Fig. 16.4b for chaotic quantum states produced in the source is predicted by the powerful van Cittert–Zernike theorem [6, 26, 50–52]. It picks out the coherent fraction of light emitted by a source, even if the source is chaotic. In classical electromagnetism, radiation (i.e. acceleration fields [53]) can only separate from the charge if it is defined at least over the dimension of its average wavelength. Hence any light-emitting source contains a coherent fraction that arises from waves emitted from the minimum coherence area of order λ2 [54, 55]. In classical optics, the 1D pattern in Fig. 16.4b arises from waves emitted from regions of lateral size  λ within the two slits that interfere at detection points. The more fundamental photon nature of light, described by QED, just happens to reduce in first order to the conventional wave formalism. One has to realize, however, that the classical theory is based on ad hoc assumptions or postulates which make it work. The fundamental origin of these assumptions, like the perceived existence of spherical waves and the validity of the Fresnel–Huygens principle, emerges in lowest order QED as a consequence of the interference of single photon probability amplitudes associated with all possible photon paths to a given detection point. The independence of the first order quantum pattern of the total number of emitted photons N , or the average number of photons 2n (which reflect the photon degeneracy parameter of the source) also reveals why the detection process did not have to be specified in the classical formulation. The diffracted “intensity” is simply calculated as the absolute value squared of the “wave field”, and the quality or statistics of the pattern improves linearly with “intensity”. Historically, this allowed the use of wave concept long before the true photon nature of light was known. In the following sections we will show that only QED can account for more sophisticated diffraction experiments carried out by changing the detection process, revealing the intrinsic limitation of the wave theory of light.

16.6 Second Order Double-Slit Diffraction Patterns states in Table 16.3, the second order diffraction patterns  For(2) the quantum P (ρ 1 , ρ 2 ) , given by (16.5), are calculated by use of the matrix elements of the four second order terms (5.15)–(5.18). The matrix elements, derived in Appendix A.10, are given in Table 16.4 for the collective states containing an average of n photons per mode (or a total of 2n) and in Table 16.5 for the states containing a total of N -photons in both modes. By use of these matrix elements the integrations in (16.6) over the finite slit widths involve the integrals (16.26) and (16.27). As for the first order cases they will give sinc-type functions that account for the finite slit widths and can be factored. The

870

16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

Table 16.4 Matrix elements of the second order normally ordered operator products, listed in the same sequence as in (5.15)–(5.18) with the collective quantum states of Table 16.1 Term Operator Oa Ocoh Odif Ocha ak† Aak† BakAakB

A

B ak† Aak† AakAakA ak† Bak† BakBakB ak† Aak† AakBakB ak† Bak† BakAakA C, D ak† Aak† AakAakB ak† Aak† BakBakB ak† Aak† BakAakA ak† Bak† BakAakB † † a Note that a a = a† a† and a a kA kB kA kB kB kA

n2

n2

n2

n2

n2

n2 n2 n2

n2 0 0

2n2 2n2 0 0

n2 n2 n2 n2

0 0 0 0

0 0 0 0

= akBakA . The order of two adjacent creation or destruction operators may be switched, and we have listed the operator with kA first. The states contain an average number of n photons per mode or a total number of 2n photons. Table 16.5 Matrix elements as in Table 16.4 but for the states containing a total number of N photons in the two modes Term

Operator O

OcohN

OentN

OnumN

OdifN

OchaN

A

ak† ak† akAakB

1 4 N (N −1) 1 4 N (N −1) 1 4 N (N −1) 1 4 N (N −1) 1 4 N (N −1) 1 4 N (N −1) 1 4 N (N −1) 1 4 N (N −1) 1 4 N (N −1)

0 1 2 N (N −1) 1 2 N (N −1)

1 2 4N 1 4 N (N −2) 1 4 N (N −2)

1 N (N −1) 4 1 N (N −1) 4 1 N (N −1) 4

1 N (N −1) 6 1 N (N −1) 3 1 N (N −1) 3

δ(N , 2)

0

0

0

δ(N , 2)

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

A B ak† ak† akAakA A A ak† ak† akBakB B B ak† ak† akBakB A A ak† ak† akAakA B B ak† ak† akAakB A A ak† ak† akBakB A B ak† ak† akAakA A B ak† ak† akAakB B B

B

C, D

The symbol δ(N , 2) means that we obtain 1 only for N = 2 and 0 otherwise.

remaining sum of exponentials reduce to cos-type expressions, representing the case of point-like slits. As an example we shall discuss below the calculation of the pattern for the collective coherent state.

16.6.1 Coherent State For the collective coherent state all matrix elements in Table 16.4 have the same value, independent of the permutations X,Y,U,V=A,B, given by coh |ak† U ak† V akX akY |coh  = n2 . They can therefore be factored and (16.6) is evaluated by use of (16.26) and (16.27) as 

     ka ka Z. ρ1 sinc2 ρ2 P(2) (ρ 1 , ρ 2 ) coh = n2 sinc2 2z 0 2z 0

(16.34)

16.6 Second Order Double-Slit Diffraction Patterns

We are left with a sum denoted of the form e

±i 2zk ρ1 0



e

871

Z consisting of sixteen exponential terms, each

±i 2zk ρ1 ±i 2zk ρ2 0

e

0

e

±i 2zk ρ2 0

.

(16.35)

The two possible signs in each term correspond to parallel or antiparallel orientations of the source coordinates rI relative to the detector coordinates ρ i when the dot products in (16.6) are evaluated. The sixteen terms are those in (5.15)–(5.18) with unit expectation values of all operators. The terms combine to cos-type terms representing the case where the two slits are points, previously treated in Sect. 5.5.2. After some algebra we obtain the final result           (2) k k ka ka ρ1 cos2 ρ2 sinc2 ρ1 sinc2 ρ2 . P (ρ 1 , ρ 2 ) coh= n2 cos2 2z 0 2z 0 2z 0 2z 0      point like slits

finite slit width

(16.36) The second order patterns for the other quantum states in Table 16.1 are derived in Appendix A.10.

16.6.2 Plots of the Second Order Patterns The calculated shapes G (2) (ρ 1 , ρ 2 ) are plotted in Fig. 16.5 in the order of the cases in Fig. 16.1. Again the shapes and scaling factors P2 are given on the right for convenience. For the same quantum states, the degeneracy still present in the first order patterns in Fig. 16.4b is now lifted. The diffraction patterns have become characteristic signatures of the different quantum states of light, revealing the new paradigm. The pattern in Fig. 16.5a is characteristic of coherent states emitted by the source. The shape is independent of the total average number of photons, 2n, and the specific photon number, N , in the substates which only determine the scaling factor P2 . The coherent states and their substates give the same patterns for the two detection schemes in Fig. 16.2c. For the specific source scheme on the right in Fig. 16.1a, the source is described by a second order collective coherent state that contains an average number of n = 2 photons per mode. The collective state consists of a Poisson substate distribution extending up to about N = 10 as shown in Fig. 16.3a. The collective state may be imaged in two ways. The two-photon coincidence detection scheme in Fig. 16.2c will pick out its representative N = 2 substate defined by the arrival of photon pairs. On the other hand, a CCD detector will integrate over all arriving numbers of photons and thus record the entire collective state, as previously conjectured [56, 57]. This is discussed in more detail in conjunction with the degree of second order coherence of the collective coherent state and its substates in the following section.

872

16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

Fig. 16.5 Second order double-slit diffraction patterns for the indicated two-mode quantum states and their substates for  = 4a. Blue and red colors represent patterns for the two detection schemes in Fig. 16.2c. a Pattern G (2) (ρ 1 , ρ 2 ) for the coherent state and its binomial substates. b Pattern for the N = 2 entangled state, and c for the N = 2 number state. d Pattern for the phase-averaged coherent state and its substates, and e for the chaotic state and its substates

The patterns of the N = 2 entangled state |φent2  given by (16.22) and number state |φnum2  expressed by (16.24) are shown in Fig. 16.5b, c. They are complete opposites, corresponding to the exchange of the red and blue detection schemes in Fig. 16.2c. In particular, the well-studied entangled state |φent2  [21, 27, 28] incorporates the essence of quantum behavior since it is maximally entangled, and it plays a prominent role in quantum information science [43–46]. The complementary behavior of the two states holds a central position in quantum optics since their diffraction patterns cannot be explained by the wave formalism. The same behavior was already encountered for a circular flat-top source as shown in Fig. 5.8. The key role of the two-photon entangled and number states is furthermore revealed by the fact that the two-photon case of all five source implementations in Fig. 16.1 may be represented as a linear combination of the two states, written in the general form |φ N =2 k,k  = a11 eiφ11 |1k |1k  + a02 eiφ02 |0k |2k  + a20 eiφ20 |2k |0k  ,

(16.37)

16.6 Second Order Double-Slit Diffraction Patterns

873

where the coefficients ai j are real and for equal occupation of the two modes we have a02 = a20 . The normalization condition furthermore links the coefficients by 2 2 2 + a20 + a02 = 1. a11 In particular, the N = 2 coherent substate (16.14), given by |φcoh2 

k,k 

   1 √    = 2|1k |1k + |0k |2k +|2k |0k 2

(16.38)

corresponds to an in-phase addition of the N = 2 state |φnum2  and the state |φent2 , identified by square brackets. Thus the coherent two-photon pattern in Fig. 16.5a arises from the interference of the number and entangled states, expressed by (A.73). The N = 2 entangled and number states are also responsible for the evolution of the patterns in Figs. 16.5c–e. The number state |φnum2  = |1k |1k  creates the diffraction fine structure shown in (c) which is superposed on a background, shown in green, in the patterns of the collective phase-diffused and chaotic states and their substates in (d) and (e). The size of the background in these patterns is determined by different relative contributions of the entangled and number states, which combine with random relative phases in the general two-photon state (16.37). The key difference of the N = 2 phase-diffused coherent substate (16.17) and the chaotic substate (16.20) is revealed by writing them respectively as    1 1 |φdif2 k,k  = √ eiϕ |1k |1k  + √ e2iϕ |0k |2k  +|2k |0k  2 2

(16.39)

   1 |φcha2 k,k  = √ eiϕ1 |1k |1k  + eiϕ2 |0k |2k  + |2k |0k  . 3

(16.40)

and

In the last expression we have rewritten (16.20) by eliminating an unimportant overall phase factor through the choice φ20 = 0, φ02 = ϕ2 , and φ11 = ϕ1 . The different contributions of the entangled substate in (16.39) and (16.40), identified by rectangular brackets, relative to the number state is the reason for the change in the green background in the patterns in Fig. 16.5c–e. The background quantitatively scales with the square of the coefficients expressing the number and entangled state contributions to these states. The entangled state reflects the simultaneous birth of two photons within a given slit. This situation is encountered in practice for incoherent or chaotic sources since the creation of only single photons per slit, reflected by a pure |1k |1k  state (pattern in Fig. 16.5c), requires special source preparation [33–36]. The dashed red envelope function of the chaotic state pattern in Fig. 16.5e is the 1D manifestation of the famous Hanbury Brown–Twiss (HBT) result [58–62], where a circular or rectangular 2D source is replaced by a 1D slit of width a. The HBT effect was first derived quantum mechanically by Fano [63] using Feynman’s concepts of probability amplitudes and played an important role in Glauber’s development of quantum optics as recalled in his Nobel lecture [64].

874

16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

In QED, the HBT arises naturally because of the structure of the chaotic quantum states. This is best revealed by the two-photon chaotic substate (16.40) which contains an entangled part of 50%. It is therefore not surprising that many quantum optics experiments, first performed with entangled biphotons, can also be performed with chaotic light. A prominent example is “ghost imaging” [21, 65] which is also possible with chaotic sources [66], albeit with reduced contrast [67]. The HBT effect may also be explained classically as arising from “intensity fluctuations” [52, 68] which of course are nothing but the fluctuations in the created number of photons per unit time and area. In the semi-classical picture, the constant background is typically attributed to two photons that “accidentally” arrive at the two detectors in coincidence. This arrival condition is assured for entangled biphotons created in parametric down conversion by their simultaneous birth at the same place. It may occur for a chaotic source when two photons born at different times and positions “accidentally” arrive at the same time because the difference in birth time is compensated by the difference in travel time (distance) [6]. From a practical or detection point of view, the entangled and “accidental” scenarios are indistinguishable. The formal statistical optics derivation utilizes the so-called Reed theorem [69] or complex Gaussian moment theorem [26, 70], which veils the underlying fundamental quantum processes.

16.6.3 Degree of Second Order Coherence The difference of the second order patterns in Fig. 16.5 is also reflected by the degree of second order coherence (16.7) of the respective quantum states, which for the double-slit geometry is expressed by  P(2) (ρ, −ρ) g (ρ, −ρ) =  2 . P(1) (ρ, ρ) 

(2)

(16.41)

It is evaluated in Appendix A.11 for the different cases in Fig. 16.5. (2) For the collective coherent state |coh , we have gcoh (ρ, −ρ) = 1 which together (1) with gcoh (ρ, −ρ) = 1 given by (16.32) is the signature of a second order coherent state [16, 71]. It is remarkable that the coherent substates |φcohN  are not second (2) (ρ, −ρ) = 1 − N1 , which approaches the coherent value order coherent since gcohN of unity only in the limit of a large number of photons in the substates (see Sect. 5.6.3.1). (2) (ρ, −ρ) = 1/2 for the two-photon coherent substate In particular, we obtain gcoh2 |φcoh2  given by (16.38). This state, describing a coherent biphoton [6, 56], is created when a single photon clones itself in a stimulated decay process. Its lack of second order coherence, which was previously not recognized [57], arises from the fact that in the presence of a single photon, an atom may also decay with equal probability via spontaneous photon emission. This is expressed by the well-known factor 1 + n,

16.6 Second Order Double-Slit Diffraction Patterns

875

where 1 is the relative probability that an excited electronic state decays spontaneously in the absence of other photons, and n is the relative probability that the decay is stimulated by the presence of n photons in the same mode. The lack of second order coherence of the N = 2 substate |φcoh2 , produced through single photon (n = 1) stimulation, is reflected by the so-called no cloning theorem [72–75]. Second order coherence and perfect cloning are only reached when many photons in the same mode cooperate to completely control excitation and de-excitation of an atom [31]. This corresponds to the case shown on the right in Fig. 16.1a where an incident temporally coherent pulse with high degeneracy parameter drives the atoms in the film to a collective second order coherent state with equal populations in the ground and excited states. The coupled atom-photon system then becomes second order coherent, and the “no cloning” theorem no longer applies. The two-photon entangled state |φent2 , given by (16.22), yields the surprising (2) (ρ, −ρ) = 1. A state is second order coherent, however, only if both g (2) result gent2 (1) and g are unity. This is not the case because g (1) is not unity according to (16.33). Instead, the state |φent2  causes the constant background in Fig. 16.5d, e as discussed in the previous section. The unit value of the background is therefore deceiving since it does not reflect second order coherence, which in the semi-classical explanation has led to its description as “accidental coincidences”. For the two-photon number state |φnum2 , the collective phase-diffused coherent state |dif  and the collective chaotic state |dif , we find g (2) (ρ, −ρ) = G (2) (ρ, −ρ). The red patterns in Fig. 16.5c–e therefore represent the degree of second order coherence of these states. For the substates |φdifN  and |φchaN  the expressions for g (2) , given by (A.108) and (A.110), are similar but contain additional prefactors. The values of the degree of second order coherence g (2) (ρ 1 , ρ 2 ) of the different quantum states complement the information revealed by the shapes of their diffracthe patterns of the collective states G (2) (ρ 1 , ρ 2 ) in Fig. 16.5. While  tion patterns (2) i |P (ρ 1 , ρ 2 )|i and their substates φiN |P(2) (ρ 1 , ρ 2 )|φiN can in principle be distinguished through their scaling factors P2 , their difference is directly revealed by the normalized degree of second order coherence g (2) (ρ 1 , ρ 2 ). Examples are the (2) (2) = 1 for the collective coherent state and gcoh2 = 1/2 for its different values of gcoh two-photon substate which exhibit the same diffraction shapes.

16.6.4 The Evolution from First to Second Order The behavior of independent photons in first order QED may also be accounted for by the wave theory, augmented by certain ad hoc recipes like the Huygens–Fresnel principle which make it work. In second order QED, phenomena arise that simply cannot be explained by the wave theory of light, clearly revealing its incompleteness. The hallmark of second order QED is the existence of correlations between photons, the most heralded phenomenon being photon entanglement over large distances [45, 46]. More precisely, the second order case covers phenomena associated with a

876

16 Quantum Diffraction: Emergence of the Quantum Substructure of Light

number of N ≥ 2 photons that arise from the correlations between their probability amplitudes. These correlations are absent in first order QED or conventional quantum mechanics, as expressed by Dirac’s famous statement. Comparison of the first order patterns in Fig. 16.4 with the second order ones in Fig. 16.5 directly reveals how the remaining degeneracy in the first order patterns is lifted in second order. In particular, the evolution leads to distinct patterns for the fundamental two-mode entangled and number states, and their central role becomes apparent. In all cases, the shapes of the second order diffraction profiles (apart from any constant background) are seen to be the square of the corresponding first order patterns. For the coherent states the effective width of the first order pattern for  ≥ 2a is given by [6] ka π z0



∞ cos

2

−∞

   k 2 ka ρ sinc ρ dρ = 1 2z 0 2z 0

(16.42)

while that of the second order coherent pattern is given by ka π z0



∞

−∞

 cos4

   1 k ka ρ sinc4 ρ dρ = . 2z 0 2z 0 2

(16.43)

Photon conservation then requires that the reduction in effective width of the second order pattern by a factor of 2 is compensated by a factor of 2 larger peak value. This illustrates that with increasing order of coherence the pattern is increasingly centered around the forward direction. When extended to higher order, this leads to the remarkable result that, in principle, an n th order coherent state no longer diffracts and the collective photon state propagates on particle-like trajectories [6].

16.7 Summary The key message of this chapter is that diffraction patterns can simply be viewed as encoded signatures of different quantum states of light. This is revealed by a quantum formulation of diffraction that goes beyond the first order description in conventional quantum mechanics. The theoretical link between quantum states and their characteristic diffraction images is revealed by modern versions of Young’s double-slit diffraction experiment, summarized in Fig. 16.1. Ironically, the very experiment that 200 years ago led to the notion that light is a wave, can therefore be used to disprove this hypothesis. We note that the true photon nature of what we call electromagnetic (EM) radiation is not restricted to the short wavelength range extending from the optical to the x-ray regime. It is a universal feature of EM radiation despite the power of Maxwell’s classical theory of electromagnetic waves. Owing to the lower energy of photons

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below the visible range, it just becomes increasingly difficult to detect them since typical detectors at room temperature have a thermal background noise corresponding to about 25 meV. Today, even microwave photons may be detected by use of photon counters based on Josephson junctions [77]. In principle, the wave theory can be abandoned altogether today. In practice, it has served us well and may continue to be used with the understanding of its limitations. The broad and difficult concept of “partial coherence” in statistical optics can be better defined in quantum optics, which is anchored in the fundamental theory of light and matter, QED. The definition of quantum coherence is more specific since it is directly linked to different quantum states of light whose interference and diffraction properties are quantified through an order-dependent degree of coherence. The quantum theory furthermore differentiates between the behavior of collective quantum states defined through statistical distributions of photons and states containing specific numbers of photons. In first order QED, the fundamental photon nature of light just happens to be describable by the ad hoc wave theory, based on spherical light waves, the magical Huygens–Fresnel principle, and the assumption that the absolute value squared of the wave field gives the diffracted intensity. The limitations and non-fundamental nature of the wave theory become apparent only in second order QED, where the concept of spherical waves needs to be replaced by an average over photon emission directions, the Huygens–Fresnel principle becomes the quantum interference of photon probability amplitudes for different source-detector paths, and the “intensity” concept is replaced by the probability of photon detection.

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Appendix

A.1 The International System of Units (SI)

Table A.1 Physical quantities, symbols, and their SI units Symbol

Physical quantity

A

Vector potential

Equivalent

V s m−1

Numerical value and units

E

Electric field

V m−1

D

Displacement

H

Magnetic field

B

Magnetic Induction

p

Electric dipole moment

m

Magnetic dipole moment

M

Magnetization

c 0

0 E

A s m−2

μμ0 H

V s m−2 (≡ T)

A m−1

Asm Vsm V s m−2 (≡ T)

Speed of light

m/V √ 1/ 0 μ0

2.998×108 m s−1

Dielectric constant

1/μ0 c2

8.854×10−12 A s V−1 m−1

μ0

Permeability

1/0 c2

4π ×10−7 V s A−1 m−1

1/4π 0

SI units prefactor

μB

Bohr magneton

eμ0 /2m e

1.165×10−29 V m s

me

Electron mass

8.99×109 V m A−1 s−1 9.109×10−31 V A s3 m−2 6.626×10−34 V A s2 = 4.136 eV fs

h

Planck’s constant



Planck’s constant

kB

Boltzmann’s constant

1.381×10−23 V A s K −1 = 8.617×10−5 eV K −1

NA

Avogadro’s number

6.02214×1023 atoms/mol

Rg

Universal gas constant

q = −e

Electron charge

e/m e

Electron charge/mass

h/2π

kB NA

1.055×10−34 V A s2 = 0.6582 eV fs

8.31446 V A s K−1 mol−1 = 5.188×1019 eV K −1 mol−1 −1.602×10−19 A s

ω/B

1.759×1011 rad s−1 T−1

(continued)

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0

881

882

Appendix

Table A.1 (continued) m e c2

0.819×10−13 V A s = 0.511 MeV

Electron rest energy

ER

Rydberg energy

a0

Bohr radius

r0

Classical electron radius

σe

Thomson cross section

αf

Fine structure constant

m e e4 /22 (4π 0 )2 4π 0 2 /m e e2 e2 /4π 0 m e c2 8πr02 /3 e2 /4π 0 c

C

Compton scattering length

/m e c

13.606 eV 0.529×10−10 m 2.818×10−15 m 0.665×10−28 m2 = 0.665 barn = 1/137.04 3.862 × 10−13 m

Conversions of units 1 Oersted (Oe) = (1000/4π ) A m−1 = 79.59 A m−1 1 Tesla (T) = 1 N A−1 m−1 = 1 V s m−2 (i.e. 1 T corresponds to 104 Oe) 1 Ohm () = 1 V A−1 1 Coulomb (C) = 1 A s 1 Newton (N) = 1 V A s m−1 1 Kilogram (kg) = 1 V A s3 m−2 1 Farad (F) = 1 A s V−1 1 Joule (J) = 1 N m = 1 V A s 1 Watt (W) = 1 V A = 1 J s−1 1 eV = 1.602 × 10−19 V A s 1 eV/kB = 1.1605 × 104 K (energy to temperature) 1 eV/ h = 2.418 × 1014 Hz (energy to cycle frequency) 1 eV/ hc = 8066 cm −1 (energy to wavenumber, also 1 cm−1 = 1 Kayser) hν[eV] = 1239.852/λ [nm] (photon energy to wavelength and vice versa) 1 μB /μ0 = 0.578 × 10−4 eV T−1 1 barn (b) = 1 × 10−28 m2 1 deg (◦ ) = π/180 rad = 17.45 mrad 1 arcmin = 1/60◦ = 290.9 µrad

A.2

Resonance Lineshapes

A.2.1 Lorentzian Lineshape and Integral

The Lorentzian lineshape function centered at x0 is L(x) = A

( /2)2 , (x − x0 )2 + ( /2)2

(A.1)

where A is the peak value and is the full width at half maximum (FWHM). For A = 2/(π ) the Lorentzian has unit area.

Appendix

883

Its integral is a step function given by x I (x) = −∞



1 1 L(u) du = H + arctan 2 π



x − x0

/2

 ,

(A.2)

where H = A π/2 is the step height.

A.2.2 Gaussian Lineshape and Integral

The Gaussian lineshape function centered at x0 is given by    2  (x − x0 )2 a (x − x0 )2 G(x) = A exp − = A exp − . 2 σ2 2 G2

(A.3)

Here A is the peak height, and σ is the rms (one standard deviation) width. The full width at half maximum √(FWHM) G is related to the√rms width according √ to G = a σ , where a = 2 ln 4 = 2.355. For A = 1/(σ 2π ) = a/( G 2π ) the Gaussian has unit area. Its integral is a step function given by 

x G(u) du = H

I (x) = −∞

  1 1 x − x0 , + erf √ 2 2 2σ

(A.4)

√ where H = A σ 2π is the step height and erf is the error function.

A.2.3 Voigt Lineshape The Voigt function is a convolution of the Lorentzian of natural line FWHM and a Gaussian of FWHM G . With the same notations as for the Lorentzian and Gauss functions above, the Voigt lineshape function is given by   2 

2 √ 0 )+ /2) Re exp − a (x−x2 0 −i /2) erfc a(i(x−x 2 2 G  2G 2  . V (x) = A a

a

exp 8 2 erfc 2√2

G

G

(A.5)

884

Appendix

Here erfc[x] = 1 − erf[x] and A is the peak height. The profile has unit area for  2 2   a

a

a exp erfc √ , (A.6) A= √ 8 G2 2π G 2 2 G and the FWHM linewidth is given by

V = 0.535 +

A.3

 0.217 2 + G2 .

(A.7)

Dirac δ-Function

The Dirac δ-function is zero for all x except at one point x = a where it is infinite δ(x − a) =

0 x = a ∞ x = a.

(A.8)

It is defined only through its integrated property, according to ∞ δ(x − a) dx = 1

(A.9)

f (x) δ(x − a) dx = f (a) .

(A.10)

−∞

and has the sifting property ∞ −∞

It is commonly assumed that the dimension of x in the argument defines the dimension of the δ function, which may also be explicitly denoted as δ (n) .

A.4

Fourier Transforms and Parseval’s Theorem

The Fourier transform formalism introduced below is based on J. W. Goodman’s book “Introduction to Fourier Transforms” [1]. More recently, the same author has written a wonderful little tutorial on the practical implementation of continuous and discrete Fourier transforms using Mathematica [2].

Appendix

885

A.4.1 1D Fourier Transform The (forward) Fourier transform of a one-dimensional function f (x) is defined as ∞ F( p) =

f (x) e− 2π i p x dx ,

(A.11)

F( p) e2π i p x d p.

(A.12)

−∞

and the inverse transform is ∞ f (x) = −∞

The Fourier relations are accompanied by a powerful theorem involving the absolute values squared of the transform pair, called Parseval’s theorem,1 sometimes called Rayleigh’s theorem or Plancherel’s theorem ∞

∞ | f (x)| dx =

|F( p)|2 d p .

2

−∞

(A.13)

−∞

In practice, one often encounters conjugate variables that are linked to the original variables by inclusion of a factor of 2π . Examples are the relations between time and the angular frequency, ω = 2π ν = 2π/t, or between wavelength and wavevector k = 2π/λ. By substituting q = 2π p into (A.11) and (A.12) we obtain the following equivalent transform formulation, which we shall use in this book.2 In general, the transforms F(q) and f (x) have different dimensions. In this book we shall use the following formulation of the 1D Fourier transform and Parseval’s theorem ∞ f (x) e− i q x dx . (A.16) F(q) = −∞

1 It is named after Marc-Antoine Parseval des Chênes and dates back to 1799 when he presented it to the Académie des Sciences in Paris. 2 Another definition often used in physics (and e.g. in Mathematica) is to split the factor of 2π between the forward and inverse transforms to make them symmetrical,

1 F(q) = √ 2π and 1 f (x) = √ 2π

∞

f (x) e− i q x dx

(A.14)

F(q) ei q x dq.

(A.15)

−∞

∞ −∞

886

Appendix

1 f (x) = 2π

∞

∞ F(q) e i q x dq.

1 | f (x)| dx = 2π

∞ |F(q)|2 dq

2

−∞

(A.17)

−∞

(A.18)

−∞

Parseval’s theorem (A.18) assures energy conservation upon Fourier transformation (see below).

A.4.2 1D Transformation Under Preservation of Dimension It is convenient to define Fourier transformed fields in the time t and angular frequency ω = 2π ν (or energy ω) domain with the same dimension, which for electric fields is expressed by [V/m] in the SI system. This requires introduction of a dimension-preserving constant C of dimension [time]. We write the temporal electric field distribution as E(t) = E 0 f (t) ,

(A.19)

where f (t) is a dimensionless temporal distribution function of unit peak value, so that E(t = 0) = E 0 . The fields in the time and (reciprocal) frequency domains can then be linked by a 1D Fourier transform according to ∞ 1 f (t) e−i ω t dt . E(ω) = E 0 C −∞  

(A.20)

F(ω)

The frequency distribution function F(ω) is also dimensionless of unit peak value, so that E(ω = 0) = E 0 , and its inverse Fourier transform reproduces E(t) according to ∞ C F(ω) ei ω t dω . E(t) = E 0 2π −∞  

f (t)

(A.21)

Appendix

887

Then Parseval’s theorem links the associated intensities according to ∞

C2 |E(t)| dt = 2π

∞ |E(ω)|2 dω

2

−∞

(A.22)

−∞

and therefore assures intensity or energy conservation upon Fourier transforming. The constant C needs to satisfy the Fourier relations as well as Parseval’s theorem. It may simply be determined by setting ω = 0 in (A.20) or alternatively at t = 0 from (A.21), which for E(t = 0) = E(ω = 0) = E 0 yields ∞ C=

−∞

E(t) dt E0

2π E 0 . −∞ E(ω) dω

= ∞

(A.23)

A.4.3 Effective 1D Distribution Widths and Transform Limit Since Parseval’s relation (A.22) assures intensity conservation, we may use it to define effective intensity distribution widths. The idea is to define the integrated intensity as its peak value E 02 times effective widths of the distribution functions in the time and frequency domains according to ∞ Effective widths : t =

−∞

|E(t)|2 dt E 02

∞ and ω =

−∞

|E(ω)|2 dω E 02

. (A.24)

The so-defined integrated intensities E 02 t and E 02 ω correspond to those under flattop intensity distributions of peak values E 02 and widths t and ω containing the same photon energy. The product of the effective widths constitutes a time-bandwidth product that determines the transform limit. It is obtained as ∞ t ω =

2 −∞ |E(t)| dt

E 02

∞

2 −∞ |E(ω)| dω

E 02

2  ∞     2π = 2 4  |E(t)|2 dt  , (A.25) C E0   −∞

where on the right we have used (A.22). Inserting the expression for C given by the temporal expression in (A.23) we obtain 2  ∞ 2   −∞ |E(t)| dt t ω = 2π  2  ∞ E 02  −∞ E(t) dt 

  K

(A.26)

888

Appendix

which expresses the transform limit or time-bandwidth product in the form (4.20) by assuming effective distribution widths and K given by (4.24) for a classical field.

A.4.4 2D Fourier Transform The 2D transform is defined in analogy to (A.16) and (A.17)  ∞ F(qx , q y ) =

f (x, y) e−i(qx x+q y y) dx dy

(A.27)

−∞

1 f (x, y) = 2 4π

 ∞ F(qx , q y ) ei(qx x+q y y) dqx dq y .

(A.28)

−∞

Parseval’s theorem for the 2D case reads  ∞  ∞ 1 2 | f (x, y)| dx dy = 2 |F(qx , q y )|2 dqx dq y . 4π −∞

(A.29)

−∞

A.4.5 2D Transformation Under Preservation of Dimension In the following we shall consider the 2D Fourier transform under preservation of dimension similar to the 1D case treated in Sect. A.4.2. While the transformation step between conjugate 1D time—energy scalars is very similar to that between 2D space-momentum vectors, in the latter case one is typically also interested in including the propagation from the source to the detector plane, which contains the measured diffraction pattern. Because of the linearity of the Fourier transform, however, the desired transformation E(r, 0) ↔ E(q, z 0 ) can be broken down into a pure transform step E(r) ↔ E(q) times a propagation step z = 0 ↔ z = z 0 . It is therefore possible to consider the transform step alone, without propagation, as we shall do below. The intensity propagation can simply be included after the transformation step by a propagation factor. For a source of area As and a detector plane at large distance z 0 the intensity propagation is given by a multiplicative factor A2s /(λ2 z 02 ) as identified, for example, in the far-field diffraction pattern (8.25). As in Sect. A.4.2 we describe the real space coherent field with coordinates r = (x, y) as E(x, y) = E 0 f (x, y) ,

(A.30)

Appendix

889

where f (x, y) is dimensionless distribution function of unit peak value. Its Fourier transform is then given in coordinates q = (qx , q y ) as 1 E(qx , q y ) = E 0 C

 ∞ f (x, y) e−i(qx x+q y y) dx dy . −∞



(A.31)



F(q x ,qy )

Here F(qx , q y ) is a dimensionless reciprocal space distribution function of unit peak value and C of dimension [length2 ] assures preservation of the field dimension. The back transformation reproduces the real space field according to C E(x, y) = E 0 4π 2

 ∞ F(qx , q y ) ei(qx x+q y y) dqx dq y . −∞



(A.32)



f (x,y)

Parseval’s theorem links the associated intensities according to  ∞  ∞ |C|2 2 |E(x, y)| dx dy = 2 |E(qx , q y )|2 dqx dq y . 4π −∞

(A.33)

−∞

The constant C needs to satisfy the Fourier relations as well as Parseval’s theorem. It may simply be determined by setting q = 0 in (A.31) or alternatively at r = 0 in (A.32) , and with E(r = 0) = E(q = 0) = E 0 we obtain  ∞ C=

−∞

E(x, y) dx dy E0

4π 2 E 0 . −∞ E(q x , q y ) dq x dq y

=  ∞

(A.34)

A.4.6 Effective 2D Distribution Areas and Diffraction Limit Since Parseval’s theorem assures energy conservation upon field transformation, we can now define effective real source areas and associated reciprocal space “areas” that determine the solid angle of emission. The effective source area of an arbitrary intensity distribution function is defined as  ∞ Effective source area : Aeff =

−∞

|E(x, y)|2 dx dy E 02

.

(A.35)

890

Appendix

The effective solid angle associated with the reciprocal space momentum transfer q is similarly defined as  ∞

−∞

Effective solid angle : deff =

|E(qx , q y )|2 dqx dq y k 2 E 02

,

(A.36)

where k = 2π/λ. The product Aeff deff then defines the diffraction limit (see Fig. 4.1) and by use of (A.33) and then inserting C from (A.34) we obtain

Aeff deff

2  ∞ 2   −∞ |E(x, y)| dx dy 2 =  2 λ .  ∞ E 02  −∞ E(x, y) dx dy 

 

(A.37)

H

This expresses the diffraction limit in the form (4.19) for effective source areas and solid emission angles with H given by (4.25) for a classical field.

A.5

Spherical Harmonics and Tensors

Following Racah, we define spherical tensors Cm(l) for l ≤ 4, in terms of the spherical harmonics as follows  4π (l) Cm = (A.38) Yl,m (θ, φ) . 2l + 1 Table A.2 gives expressions for Cm(l) for l = 0, 1, 2, 3, 4. (l)

Table A.2 Racah tensor operators Cm for l = 0, 1, 2, 3, 4 defined in (A.38), expressed in Cartesian and spherical coordinates, according to x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ (0) C0 = 1 (1)

C0 = cos θ (1) C±1 (2) C0

=

z r

= ∓ √1 sin θ e±iφ

= ∓ √1

=

1 2

=

(x±i y) r 2 1 (3z 2 −r 2 ) 2 r2

C0 =

1 2

=

1 (5z 2 −3r 2 )z 2 r3

(2) C±1 (2) C±2 (3)

2

(3 cos2 θ − 1)  3 cos θ sin θ e±iφ = ∓ 2  = 38 sin2 θ e±2iφ (5 cos3 θ − 3 cos θ)

 y)z = ∓ 23 (x±i r2  y)2 = 38 (x±i r2 (continued)

Appendix

891

Table A.2 (continued) (3)

3 sin θ (5 cos2 θ − 1) e±iφ C±1 = ∓ 16  (3) 2 ±2iφ C±2 = 15 8 cos θ sin θ e  (3) 5 C±3 = ∓ 16 sin3 θ e±3iφ (4)

C0 = (4) C±1 (4) C±2 (4) C±3 (4) C±4

1 (35z 4 −30z 2 r 2 +3r 4 ) 8  r4 5 (x±i y)(7z 3 −3zr 2 ) = ∓ 16 r4  5 (x±i y)2 (7z 2 −r 2 ) = 32 r4  z(x±i y)3 = ∓ 35 r4  16 35 (x±i y)4 = 128 r4 (l) (l) property (Cm )∗ = (−1)m C−m

(35 cos4 θ − 30 cos2 θ + 3)  5 = ∓ 16 sin θ (7 cos3 θ − 3 cos θ) e±iφ  5 = 32 sin2 θ (7 cos2 θ − 1) e±2iφ  = ∓ 35 cos θ sin3 θ e±3iφ  16 35 = 128 sin4 θ e±4iφ 1 8

The operators have the complex conjugate

 3 (x±i y)(5z 2 −r 2 ) = ∓ 16 r3  15 z(x ± i y)2 = r3 8 5 (x±i y)3 = ∓ 16 r 3

=

(1) A.5.1 Relations between First and Second Order Tensors C m (2) and C m

Table A.3 Relationships of Racah tensor operators Cm(1) and Cm(2) derived from the expressions listed in Table A.2 (2)

(1)

(1)

(1)

(1)

C0 = C0 C0 + C−1 C1 √ (2) (1) (1) C±1 = 3 C0 C±1  (2) (1) (1) C±2 = 23 C±1 C±1

(l)

(l)

The operators have the complex conjugate property (Cm )∗ = (−1)m C−m , and they commute (l) (k) (k) (l) Cm Cq = Cq Cm .

A.6

s, p, and d Orbitals

Table A.4 gives the often used real s, p, and d orbitals in terms of linear combinations of the (complex) spherical harmonics.

892

Appendix

Table A.4 Mathematical description of s, p, and d orbitals in terms of spherical harmonics Yl,m l = |l, m s = √1 = Y0,0 4π 3 x px = 4π r = √1 ( Y1,−1 − Y1,+1 ) 2  3 y √i ( Y1,−1 + Y1,+1 ) py = 4π = 2  r 3 z pz = 4π = Y 1,0 r  15 x y dx y = 4π = √i (Y2,−2 − Y2,+2 ) 2 2  r 15 x z √1 (Y2,−1 − Y2,+1 ) dx z = 4π = 2 2  r 15 yz √i (Y2,−1 + Y2,+1 ) d yz = 4π = 2 2  r 2 2 15 (x −y ) 1 √ dx 2 −y 2 = 16π = (Y + Y2,+2 ) r2 2 2,−2  5 (3z 2 −r 2 ) d3z 2 −r 2 = 16π r 2 =Y2,0 The orbitals oi are real and normalized according to



|oi |2 d =

2π  π

|oi |2 sin θ dθ dφ = oi |oi  =

00

1.

A.7

Spin-Orbit Basis Functions and Matrix Elements

Table A.5 lists one-electron spin-orbit coupled functions |l, s, j, m j  for s, p, and d electrons in terms of the uncoupled basis functions |l, s, m l , m s . The given transformation is a specific example of the more general transformation between functions in different coupling schemes given by the Clebsch–Gordon coefficients [3–5]. In Tables A.6 and A.7 we list the action of the spin and angular momentum operators in a Cartesian coordinate system.

Appendix

893

Table A.5 Spin-orbit wave functions for s, p and d orbitals. The electron spin wavefunctions are |s, 1/2 = ↑ (spin-up) and = |s, −1/2 =↓ (spin-down) One-elec. Config. label |l, s, j, m j  basis |l, m l , s, m s  basis label 2S+1 L lj j mj Yl,m l ↑ or ↓ J s1 2

2S

p1

2P

p3

2P

2

2

1 2

+ 21

Y0,0 ↑

1 2

1 2

− 21 + 21

Y0,0 ↓ √ √1 (−Y1,0 ↑ + 2 Y1,+1 ↓) 3 √ √1 (− 2 Y1,−1↑ +Y1,0↓)

3 2

3 2

1 2

− 21 + 23 + 21 − 21

d3 2

2D

3 2

3 2

− 23 + 23 + 21 − 21 − 23

d5 2

2D

5 2

5 2

+ 25 + 23 + 21 − 21 − 23 − 25

3

Y1,+1 ↑ √ √1 ( 2 Y1,0 ↑ +Y1,+1↓) 3 √ √1 (Y1,−1↑ + 2 Y1,0↓) 3

Y1,−1↓ √1 (−Y2,+1 ↑ +2 Y2,+2 ↓) 5 √ √ √1 (− 2 Y2,0↑ + 3 Y2,+1↓) 5 √ √ √1 (− 3 Y2,−1↑ + 2 Y2,0↓) 5 √1 (−2 Y2,−2↑ 5

+Y2,−1↓)

Y2,+2↑ √1 (2 Y2,+1↑ +Y2,+2↓) 5 √ √ √1 ( 3 Y2,0↑ + 2 Y2,+1↓) 5 √ √ √1 ( 2 Y2,−1↑ + 3 Y2,0↓) 5 √1 (Y2,−2↑ +2 Y2,−1↓) 5

Y2,−2↓

The configuration label is either for one-electron or one-hole occupation in the particular subshell. Table A.6 Spin operators sα (α = x, y, z) acting on spin states sz = +1/2 =↑ and sz = −1/2 =↓, where we have assumed the quantization axis z sx | ↑ = 21 | ↓ sx | ↓ = 21 | ↑

s y | ↑ = 2i | ↓ s y | ↓ = − 2i | ↑

sz | ↑ = 21 | ↑ sz | ↓ = − 21 | ↓

Eigenvalues are in units of . Table A.7 Angular momentum operators Lα (α = x, y, or z) acting on d orbitals, taken from Ballhausen [6] √ Lx dx z =−i dx y L y dx z = i dx 2 −y 2 − i 3 d3z 2 −r 2 Lz dx z = i d yz √ Lx d yz = i 3 d3z 2 −r 2 + i dx 2 −y 2 L y d yz = i dx y Lz d yz = −i dx z Lx dx y = i dx z L y dx y = −i d yz Lz dx y = −i 2 dx 2 −y 2 Lx dx 2 −y 2 = −i d yz L y dx 2 −y 2 = −i dx z Lz dx 2 −y 2 = i 2 dx y √ √ Lx d3z 2 −r 2 = −i 3 d yz L y d3z 2 −r 2 = i 3 dx z Lz d3z 2 −r 2 = 0 Because all matrix elements between orbitals oi are purely imaginary we have oi |Lα |on  = − oi |Lα |on ∗ = − on |L†α |oi  = − on |Lα |oi . Eigenvalues are in units of .

894

A.8

Appendix

Matrix Elements of Spherical Tensors

Below we give tables of selected matrix elements of Racah’s spherical tensors. The matrix elements obey the general relationship (k) |L , M . L , M|Cq(k) |l, m∗ = L , M|Cq(k) |l, m = (−1)q l, m|C−q

(A.39)

Matrix elements of spherical tensor operators can be conveniently obtained by use of the Wigner-Eckart theorem [5, 7, 8]. One can either use the analytical form of the Clebsch–Gordon coefficients given by Condon and Shortley [3] or Slater [4], or use modern computer programs such as Mathematica. By use of the notation of Slater’s Table 20-5 for the CG coefficients we have (S) |L , M L  = J, M |C M S

J ||C (S) ||L  (L S M L M S |L S J M ) √ 2J + 1

(S) |J, M  . = (−1) MS L , M L |C−M S

(A.40)

In Table A.8 we tabulate some important reduced matrix elements L  || C (k) ||L . A complete listing is given by Cowan [7]. Matrix elements L , M|Cq(1) |l, m  are listed in Table 10.1. Table A.8 Selected values for the reduced matrix elements L  || C (k) ||L  for p, d, and f electrons √ √ √ p||C (0) || p  = 3 d||C (0) ||d = 5 f ||C (0) || f  = 7 √ √ p||C (1) ||s = 1 d||C (1) || p  = 2 f ||C (1) ||d  = 3 3 (2) (2) d||C ||s = 1 f ||C || p  = √ 5    6 28 (2) p||C || p  = − 5 d||C (2) ||d  = − 10 f ||C (2) || f  = − 15 7 k (k)  For s electrons we have s|| C (0) ||s  = 1, and we have L  ||C (k) √ ||L = (−1) L||C ||L  [7]. In particular, we have L + 1||C (1) ||L = − L||C (1) ||L + 1 = L + 1 [5].

Appendix

895

A.8.1 Polarization Dependent p → d Transition Probabilies In Fig. A.8 we have assumed that the degeneracy in m j of the s–o split p3/2 and p1/2 core states is lifted by application of an external H field along z. H acts on both the spin and orbital parts causing an inversion of the sign of the Zeeman components m j of the 2 p3/2 and 2 p1/2 states. For an exchange field which only acts on the spin, the m j substates would have the same order of signs [10].

(1) (2) A.8.2 Sum Rules For Matrix Elements of C m and C m

Let φ L denote one of the p (L = 1) or d (L = 2) functions listed in Table A.4. We are interested in converting the XNLD cross section (11.2), illustrated in a molecular orbital picture in Fig. 11.2, into the quadrupolar form (11.8), illustrated in Fig. 11.5. This requires expressing the transition matrix elements given by the first order spherical tensors C (1) in terms of those of C (2) by use of Table A.3 and then to those of the

Polarization dependent pj to d transition probabilities dx 2–y 2 = – 1/2 (|2,+2 + |2,–2 d3z 2–r2 = |2, 0 dyz = i – 1/2 (|2,+1 + |2,–1 dxz = – 1/2 (|2,–1 – |2,+1 dxy = i – 1/2 (|2,–2 – |2,+2

+

-

+

0

18 6 2 6 6 6 9 16 3 6 6 18 6 9 3

50

30

40

3

-

0

4 3 12 8 6 3 12 3 6

10

30

20

mj |1, 1

p3/2

– 2/3 |1, 0 – 1/3 |1,-1

+3/2 +1/2 -1/2 -3/2

– 2/3 |1,-1 p1/2 - – - 1/3 |1, 0

mj -1/2 +1/2

Fig. A.8 Polarization dependent transition probabilities, all multiplied by a factor of 90, from spinorbit and exchange split p core states | j, m j  to spin-up (m s = +1/2) d valence orbitals. We have chosen the magnetic quantization axis m  z and assumed k z for circular polarization ( p = ±) and E  z for linear polarization ( p = 0). The dipole transition operators are given according to (1) (1) (1) Table 11.1 by Pz+ /r = C+1 (labeled +) and Pz− /r = C−1 (labeled −) and Pz0 /r = C0 (labeled 0). We have assumed that the degeneracy in m j of the s–o split p3/2 and p1/2 core states is lifted by application of an external H field along z. We have also given the spin-up components |1, m (m = 0, ±1) of the m j core states according to Table A.5. The transition probabilities to the spindown (m s = −1/2) d states are obtained by exchanging the circular polarization labels + and −

896

Appendix

quadrupole operators by use of (11.7). By also utilizing the sum rules in Sect. 10.4.7, we here give the key steps in doing so. The matrix elements of the first order spherical tensors obey the conjugation symmetry property, operators (1) |L M . L M|Cq(1) |lm∗ = L M|Cq(1) |lm = (−1)q lm|C−q

(A.41)

One key trick in linking the matrix  elements of the tensors C (2) and C (1) is the insertion of a complete set of states lm |lm lm| = 1, called a closure relation. This leads to the link  1 1 (1) | lm|C0(1) |φ L |2 − | lm|C1(1) |φ L |2 − | lm|C−1 |φ L |2 . 2 2 l,m (A.42) One then simplifies the sum over final states |l, m in (A.42). Although the matrix elements involve a complete set of states |lm, in practice, all matrix elements are zero except for the l = L ± 1 final states connected by the dipole operators Cq(1) . One can further simplify by expressing the terms for l = L + 1 by those for l = L − 1 through the matrix elements in Table 10.1. Let us assume we are interested in the properties of the 3d valence shell so that φ L with L = 2 is one of the d orbitals. The sum in (A.42) then extends over l = L + 1 = 3 ( f shell) and l = L − 1 = 1 ( p-shell). Since for L-edge studies we are only interested in the in the d → p matrix elements which are just the conjugate p → d matrix elements according to (A.41), we need to eliminate the f shell terms. It can be proven that for any state |L M one can link the terms for l = L + 1 and l = L − 1 in (A.42) according to φ L |C0(2) |φ L  =



| l

m|C0(1) |L

l=L+1 m

=

1 1 (1) M| − | l m|C1(1) |L M|2 − | l m|C−1 |L M|2 2 2



2

 2L − 1  1 1 (1) |L M|2 . | l m|C0(1) |L M|2 − | l m|C1(1) |L M|2 − | l m|C−1 2L + 3 l=L−1 2 2 m

(A.43) This important results allow us to evaluate the entire sum in (A.42) in terms of states l = L − 1, only. This leads to the expression 

| lm|C0(1) |φ L |2 =

l=L−1 m

By use of the identity

| L − 1||C (1) ||L|2 2L + 3 + φ L |C0(2) |φ L  . 3(2L + 1) 6L + 3

(A.44)

Appendix

897 (1) L , M − 1|C−1 |L + 1, M L , M + 1|C1(1) |L + 1, M

=

2L−1 L , 2L+3

(1) M − 1|C−1 |L − 1, M L , M + 1|C1(1) |L − 1, M

(A.45)

we furthermore have 1 (1) ∓ C1(1) |φ L |2 | lm |C−1 2 l=L−1 m

A.9

       (2) 3 (2) L 2L + 3 (2)  = − φ L C0 ∓ C2 +C−2 φ L . 6L + 3 2(6L + 3) 2

(A.46)

Quantum States and Diffraction Patterns

Here we give details of the calculations of the matrix elements leading to the first order diffraction patterns in Sect. 16.5. We here use the shorter and more convenient notation k = kA and k = kB .

A.9.1 Coherent State For the collective coherent state (16.13) the non-vanishing matrix elements are given by coh |ak† ak |coh  = coh |ak†  ak |coh  = coh |ak† ak |coh  = coh |ak†  ak |coh  ∞ N  |α|2N  m = = |α|2 . 2|α|2 m!(N − m)! e m=0 N =0

(A.47)

The one-photon detection probability is obtained as    (1)    P (x1 , x2 ) coh = coh P(1) (x1 , x2 ) coh 1  = |α|2 eik·(x2 −x1 ) + |α|2 eik ·(x2 −x1 )   4 coh |X|coh 

   + |α|2 ei(k ·x2 −k·x1 ) + |α|2 ei(k·x2 −k ·x1 ) .

 

(A.48)

coh |Y|coh 

We can now express the coordinates (k, x) in terms of (r, ρ) in Fig. 16.2a and by assuming that the detector plane is at a large distance z 0 from the source we have the

898

Appendix

relations (4.127) or kX ·x j  kz 0 −

k r X ·ρ j , z0

(A.49)

where X = A, B and kA = k and kB = k and j = 1, 2. For point sources located at r A = −r B , separated by  = |r A − r B |, and ρ 2  ρ 1 we then obtain 



   k k ρ1 cos ρ2 . P (ρ 1 , ρ 2 ) coh = |α| cos 2z 0 2z 0 

(1)

2

(A.50)

The case of finite-size double-slit sources of widths a is evaluated by integrating over all points in the slits. This yields an additional envelope function and with |α|2 = n we obtain 

     k k ρ1 cos ρ2 P(1) (ρ 1 , ρ 2 ) coh = n cos 2z 0 2z 0     ka ka × sinc ρ1 sinc ρ2 . 2z 0 2z 0

(A.51)

A.9.2 N-Photon Substate of Coherent State The diffraction pattern of the two-mode coherent substate (16.14) is calculated by use of the matrix elements φcohN |ak† ak |φcohN  = φcohN |ak†  ak |φcohN  = φcohN |ak†  ak |φcohN  = φcohN |ak† ak |φcohN  =

N 1  n N! N = . N 2 n=0 n! (N −n)! 2

(A.52)

  The diffraction pattern φcohN |P(1) (ρ 1 , ρ 2 )|φcohN is hence the collective coherent state result (A.51) with n replaced by N /2, i.e.       (1) N k k ρ1 cos ρ2 P (ρ 1 , ρ 2 ) cohN = cos 2 2z 0 2z 0     ka ka × sinc ρ1 sinc ρ2 . 2z 0 2z 0

(A.53)

When the pattern is summed over N with the proper weight coefficients, we see from the relation

Appendix

899 ∞  N α 2 |c | = n 2 N N =0

(A.54)

that the coherent result (A.51) is obtained.

A.9.3 Phase-Diffused Coherent State For the phase-diffused coherent state |dif  the non-vanishing matrix elements are given by dif |ak† ak |dif  = dif |ak†  ak |dif  =

∞ N  |α|2N  m = |α|2 2|α|2 m!(N − m)! e m=0 N =0

(A.55)

and dif |ak† ak |dif  = |α|2 eiϕ dif |ak†  ak |dif  = |α|2 e−iϕ .

(A.56)

The Y-term vanishes for a phase average over ϕ. For our assumed geometry in Fig. 1 (b) we obtain in the (r, ρ) coordinates and by integration over the slit width a and with n = |α|2     n ka k P (ρ 1 , ρ 2 ) dif = |ρ 1 − ρ 2 | sinc |ρ 1 − ρ 2 | . cos 2 2z 0 2z 0



(1)



(A.57)

A.9.4 N-Photon Substate of Phase-Diffused Coherent State For the substate |φdifN  of the phase-diffused coherent state, the matrix elements are given by (A.55) and (A.56) with |α|2 replaced by N /2. The pattern is obtained as 

     N ka k |ρ 1 − ρ 2 | sinc |ρ 1 − ρ 2 | . P(1) (ρ 1 , ρ 2 ) difN = cos 4 2z 0 2z 0

(A.58)

When summed over N with the weight factor |cαN |2 , it is seen from the relation ∞  N =0

|cαN |2

N n = |α|2 = 4 2

(A.59)

900

Appendix

that (A.58) becomes the pattern of the phase-diffused state given by (A.57), as required.

A.9.5 Chaotic State The non-vanishing matrix elements of the two-mode chaotic state |cha  are given by cha |ak† ak |cha  = cha |ak† ak  |cha  =

∞ 

N n N = n (1+ n) N +1 N =0

(A.60)

since all other matrix element contain phase factors that average to zero. The detection probability (1) is therefore the same as for the phase-diffused coherent state, and we have       (1) n ka k |ρ 1 − ρ 2 | sinc |ρ 1 − ρ 2 | . (A.61) cos P (ρ 1 , ρ 2 ) cha = 2 2z 0 2z 0

A.9.6 N-Photon Substate of Chaotic State For the chaotic substate |φchaN  the matrix elements are given by (A.60) with n replaced by N /2. The pattern is obtained as     N ka k |ρ 1 − ρ 2 | sinc |ρ 1 − ρ 2 | . P (ρ 1 , ρ 2 ) chaN = cos 4 2z 0 2z 0



(1)



(A.62)

β

If we sum over N with the weight factors |c N |2 , we obtain by use of ∞  N =0

β

|c N |2

N n = 4 2

the chaotic result (A.61).

A.9.7 N-Photon Entangled (NOON) State For the two-mode N -photon entangled state |φentN , the matrix elements are

(A.63)

Appendix

901

φentN |ak† ak |φentN  = φentN |ak†  ak |φentN  = 0 N . φentN |ak† ak |φentN  = φentN |ak†  ak |φentN  = 2

(A.64)

We obtain 

    N ka k P (ρ 1 , ρ 2 ) entN = cos |ρ 1 − ρ 2 | sinc |ρ 1 − ρ 2 | . 4 2z 0 2z 0 (1)



(A.65)

A.9.8 N-Photon Number State For the two-mode N -photon number state |φnumN  with N ≥ 2 the matrix elements are evaluated as φnumN |ak† ak |φnumN = φnumN |ak†  ak |φnumN  = 0 N φnumN |ak† ak |φnumN = φnumN |ak†  ak |φnumN  = 2

(A.66)

which is the same as for the entangled state (A.65).

A.10

Matrix Element of Second Order Coherence Operators

Here we give details of the calculations of the matrix elements leading to the second order diffraction patterns in Sect. 16.6. Again we use the notation k = kA and k = kB .

A.10.1 Coherent State For the collective coherent state (16.13), the relevant matrix elements for term A are given by coh |ak† ak†  ak ak |coh  = coh |ak†  ak† ak ak |coh  = e−2|α|

2

∞  N =0

For term B we have

|α|2N

N  n(N −n) = |α|4 . n! (N − n)! n=0

(A.67)

902

Appendix

coh |ak† ak† ak ak |coh  = coh |ak†  ak†  ak ak |coh  = e−2|α|

2

∞ 

|α|2N

N =0

N  n(n − 1) = |α|4 n! (N − n)! n=0

(A.68)

and coh |ak† ak† ak ak |coh  = coh |ak†  ak†  ak ak |coh  = |α|4 .

(A.69)

Similarly we obtain the matrix elements for terms C and D coh |ak† ak† ak ak |coh  = coh |ak† ak†  ak ak |coh  coh |ak†  ak† ak ak |coh  = coh |ak†  ak†  ak ak |coh  = |α|4 .

(A.70)

The second order detection probability becomes  (2)  1 |α|4 |α|4 P (x1 , x2 ) coh = + cos[(k − k ) · (x1 − x2 )] 4 2 2   coh |A|coh /4

|α| |α| −i(k−k )·(x1 +x2 ) |α|4 i(k−k )·(x1 +x2 ) + + + e e 4 4

2   4

4

coh |B|coh /4

|α| −i(k−k )·x1 |α|4 i(k−k )·x1 + + e e

2  2  4

coh |C|coh /4

 |α| −i(k−k )·x2 |α|4 i(k−k )·x2 + + e e ,

2  2  4

(A.71)

coh |D|coh /4

where we have identified the origin of the four terms by underbrackets. By use 2 of |α|2 = n  a+band  the identities 1 + cos[x] = 2 cos [x/2] and cos a + cos b = 2 cos  a−b cos 2 this becomes 2     n2 2 1  2 1  cos (k−k )·(x1 −x2 ) +cos (k−k )·(x1 +x2 ) P (x1 , x2 ) coh= 4 2 2     1 1   +2 cos (k−k )·(x1 −x2 ) cos (k−k )·(x1 +x2 ) . (A.72) 2 2



(2)



The pattern is seen to become symmetrical and the last term is just the interference term of the amplitudes associated with the first two terms according to  (A.73) C + D = 2 A B

Appendix

903

Converting to the (r, ρ) coordinates, we obtain for the slit separation  = |rA − rB | and detector positions ρ 2  ρ 1 



(2)



P (ρ 1 , ρ 2 ) coh = n cos 2

2

   k 2 k ρ1 cos ρ2 . 2z 0 2z 0

(A.74)

For the case of two slits of width a we obtain           (2) 2 2 k 2 k 2 ka 2 ka ρ1 cos ρ2 sinc ρ1 sinc ρ2 . P (ρ 1 , ρ 2 ) coh= n cos 2z 0 2z 0 2z 0 2z 0 (A.75) The detection probability now factors into separate symmetric contributions from points ρ 1 and ρ 2 , and the same pattern is observed for both detection schemes.

A.10.2 N-Photon Substate of Coherent State Similarly, the coherent substate (16.14) gives equal contributions from all terms A − D expressed by the matrix elements φcohN |ak† ak†  ak ak |φcohN  = φcohN |ak† ak† ak ak |φcohN  = φcohN |ak† ak† ak ak |φcohN  = φcohN |ak† ak† ak ak |φcohN  =

N 1  N! 1 n(n − 1) = N (N − 1) N 2 n=0 n! (N −n)! 4

(A.76)

and the same when exchanging k and k  . The pattern therefore is the same as for the coherent state with the substitution |α|4 = n2 by N (N − 1)/4, i.e. 

    N (N −1) 2 k 2 k ρ1 cos ρ2 cos P (ρ 1 , ρ 2 ) coh2= 4 2z 0 2z 0     2 ka 2 ka ×sinc ρ1 sinc ρ2 . 2z 0 2z 0 (2)



(A.77)

When the pattern is summed over N with the proper weight coefficients, we see from the relation ∞  N (N − 1) α 2 |c N | = n2 4 N =0

that the coherent result (A.75) is obtained.

(A.78)

904

Appendix

A.10.3 Phase-Diffused Coherent State For the phase-diffused coherent state |dif  the matrix elements for term A are obtained as dif |ak† ak†  ak ak |dif  = dif |ak†  ak† ak ak |dif  =

∞ N  |α|2N  m(m − 1) = |α|4 . 2|α|2 m!(N − m)! e m=0 N =0

(A.79)

For the first two terms in B we have dif |ak† ak† ak ak |dif  = dif |ak†  ak†  ak ak |dif  = |α|4 ,

(A.80)

while for the last two terms in B we obtain dif |ak† ak† ak ak |dif  = e2iϕ |α|4 dif |ak†  ak†  ak ak |dif  = e−2iϕ |α|4 .

(A.81)

The terms C and B are evaluated as dif |ak† ak† ak ak |dif  = dif |ak† ak†  ak ak |dif  = eiϕ |α|4 dif |ak†  ak† ak ak |dif  = dif |ak†  ak†  ak ak |dif  = e−iϕ |α|4 .

(A.82)

All terms containing the phase ϕ vanish upon phase averaging, and we obtain with |α|2 = n 

 n2 P(2) (x1 , x2 ) dif = 4



  1 1 +cos2 (k−k ) · (x1 −x2 ) . 2 2 (A.83)

When expressed in the coordinates (r, ρ) and integrated over the slit width a, we obtain       (2) n2 1 k ka |ρ 1 −ρ 2 | sinc2 |ρ 1 −ρ 2 | . (A.84) +cos2 P (ρ 1 , ρ 2 ) dif = 4 2 2z 0 2z 0   The pattern is constant, P(2) (ρ 1 , ρ 2 ) dif = 3 n2 /8 for the detection geometry ρ 1 = ρ 2 , and for ρ 1 = −ρ 2 the diffraction fine structure sits on a constant background n2 /8.

Appendix

905

A.10.4 N-Photon Substate of Phase-Diffused Coherent State For the substate |φdifN  the matrix elements are the same as those of the collective parent state with |α|4 replaced by N (N − 1)/4 so that the pattern is 

    N (N − 1) 1 2 k 2 ka |ρ 1 −ρ 2 | sinc |ρ 1 −ρ 2 | . +cos P (ρ 1 , ρ 2 ) difN = 16 2 2z 0 2z 0 (A.85) (2)



When summed over N with the weight factors |cαN |2 , it is seen from the relation ∞  N (N − 1) α 2 n2 |c N | = 16 4 N =0

(A.86)

that (A.85) becomes the pattern of the phase-diffused state given by (A.84), as required.

A.10.5 Chaotic State For the two-mode chaotic state |cha  the terms in A are evaluated as cha |ak† ak†  ak ak |cha  = cha |ak†  ak† ak ak |cha  2  ∞ n N N = (1+ n) N +1 N =0 =

N ∞  

m(N − m)

N =0 m=0

n N = n2 . (1+ n) N +2

(A.87)

Similarly, the contributions of the first two terms of B are obtained as cha |ak† ak† ak ak |cha  = cha |ak†  ak†  ak ak |cha  =

N ∞   N =0 m=0

m(m − 1)

n N = 2 n2 (1+ n) N +2

(A.88)

while the contributions from the other two terms in B average to zero, i.e. cha |ak† ak† ak ak |cha  = cha |ak†  ak†  ak ak |cha  = 0 . (A.89)

906

Appendix

The contribution by the term B is therefore only a constant resulting in a background. The matrix elements associated with terms C and D are zero owing to the fact that they contain unpaired raising and lowering operators, i.e. C + D = 0. Evaluation of detection probability and phase averaging yields   n2 2 1  (k−k ) · (x1 −x2 ) . 1+cos P (x1 , x2 ) cha = 4 2



(2)



(A.90) In the coordinates (r, ρ) and integration over the slit width a we obtain      (2)  n2 k ka P (ρ 1 , ρ 2 ) cha = |ρ 1 −ρ 2 | sinc2 |ρ 1 −ρ 2 | . 1+cos2 4 2z 0 2z 0

(A.91)

A.10.6 N-Photon Substate of Chaotic State For the chaotic substate |φchaN  the matrix elements are those of the collective parent state with n2 replaced by N (N −1)/6 and we obtain     N (N −1) 2 k 2 ka |ρ 1 −ρ 2 | sinc |ρ 1 −ρ 2 | . 1+cos P (ρ 1 , ρ 2 ) chaN= 24 2z 0 2z 0 (A.92)



(2)



β

By summing (A.92) over N with the weight factors |c N |2 we obtain by use of the relation ∞  N =0

β

|c N |2

N (N −1) n2 = 24 4

(A.93)

the chaotic result (A.91), as required.

A.10.7 N-Photon Entangled (NOON) State For the two-modeN -photon entangled state |φentN  the matrix elements for A vanish φentN |ak† ak†  ak ak |φentN  = 0 . The matrix elements for the first two terms in B are

(A.94)

Appendix

907

φentN |ak† ak† ak ak |φentN  = φentN |ak†  ak†  ak ak |φentN  N (N − 1) = 2

(A.95)

while those for the second two terms in B are non-zero only for N = 2, φentN |ak† ak† ak ak |φentN  = φentN |ak†  ak†  ak ak |φentN  = δ(N , 2)

(A.96)

where δ(N , 2) = 1 for N = 2 and zero otherwise.   The diffraction pattern is a constant for N > 2, given by P(2) (x1 , x2 ) entN = N (N −1) . For the specific N = 2 case we obtain 16 

  1 1 2 1  φent2 |B|φent2  = cos (k−k ) · (x1 +x2 ) . P (x1 , x2 ) ent2= 16 4 2 

(2)

(A.97)

In the coordinates (r, ρ) and integrated over the slit width a the pattern becomes 

    1 2 k 2 ka |ρ 1 +ρ 2 | sinc |ρ 1 +ρ 2 | . P (ρ 1 , ρ 2 ) ent2 = cos 4 2z 0 2z 0 (2)



(A.98)

A.10.8 N-Photon Number State For the two-mode N -photon number state |φnumN  the matrix elements of A are given by φnumN |ak† ak†  ak ak |φnumN  =

N2 4

(A.99)

and for the first two terms in B we have φnumN |ak† ak† ak ak |φnumN  = φnumN |ak†  ak†  ak ak |φnumN  N (N − 2) . = 4

(A.100)

Those for the second two terms in B and those of C and D vanish. We obtain in general

908

Appendix



 1 φnum2 |A + B|φnum2  P(2) (x1 , x2 ) numN = 16    N 1 2N cos2 (k − k ) · (x1 − x2 ) +(N −2) . (A.101) = 32 2

For the case N = 2 double-slit case we obtain       (2) 1 2 k 2 ka |ρ 1 −ρ 2 | sinc |ρ 1 −ρ 2 | . P (ρ 1 , ρ 2 ) num2 = cos 4 2z 0 2z 0

A.11

(A.102)

Evaluation of the Degree of Second Order Coherence

The degree of second order coherence may be obtained by use of the first order matrix elements in Tables 16.2 and 16.3 and the second order matrix elements in Tables 16.4 and 16.5 or directly from the diffractions patterns. For the collective coherent state |coh , the numerator in (16.41) factors into the denominator so that (2) (ρ, −ρ) = 1 . gcoh

(A.103)

For the coherent substates |φcohN , we obtain from (A.53) and (A.77) (2) (ρ, −ρ) = 1 − gcohN

1 . N

(A.104)

For the two-photon entangled state |φent2  given by (16.22), we obtain from (A.65) and (A.98) (2) (ρ, −ρ) = 1 . gent2

(A.105)

For number state    |φnum2  given by (16.24), we obtain from (A.102)  the two-photon and P(1) (ρ, ρ) num2 = P(1) (ρ, ρ) ent2 (2) gnum2 (ρ, −ρ) = cos2



   k ka ρ sinc2 ρ . z0 z0

(A.106)

For the collective phase-diffused coherent state |dif , we obtain (2) (ρ, −ρ) gdif

    1 2 k 2 ka = + cos ρ sinc ρ 2 z0 z0

and for the substates |φdifN  we have

(A.107)

Appendix

909

      1 1 k ka (2) gdifN (ρ, −ρ) = 1− ρ sinc2 ρ . +cos2 N 2 z0 z0

(A.108)

For the collective chaotic state |cha , we obtain (2) (ρ, −ρ) gcha

 = 1 + cos

2

   k 2 ka ρ sinc ρ z0 z0

(A.109)

and for the substates |φchaN  we have (2) gchaN (ρ, −ρ) =

      2 1 k ka ρ sinc2 ρ . 1− 1+cos2 3 N z0 z0

(A.110)

References 1. J.W. Goodman, Introduction to Fourier Optics, 4th edn. (W. H. Freeman, New York, 2017) 2. J.W. Goodman, Fourier Transforms Using Mathematica (SPIE Press, Bellingham, WA, 2020) 3. E.U. Condon, G.H. Shortley, The Theory of Atomic Spectra (Cambridge University Press, Cambridge, 1963) 4. J.C. Slater, Quantum Theory of Atomic Structure, vol. II (McGraw-Hill, New York, 1960) 5. B.W. Shore, D.H. Menzel, Principles of Atomic Spectra (Wiley, New York, 1968) 6. C.J. Ballhausen, Molecular Electronic Structures of Transition Metal Complexes (McGrawHill, New York, 1979) 7. R.D. Cowan, The Theory of Atomic Structure and Spectra (University of California Press, Berkeley, 1981) 8. J.J. Sakurai, Modern Quantum Mechanics, Revised (Addison-Wesley, Reading, Mass, 1994) 9. See Ref. [3] page 178 and Ref. [4], Sections 22.2, 25.1 and Appendix 20 10. J. Stöhr, H.C. Siegmann, Magnetism: From Fundamentals to Nanoscale Dynamics (Springer, Heidelberg, 2006)

Index

A Absorption, classical, 299 Absorption length, 332 Absorption, link to resonant scattering, 299 Acceleration fields, 67, 85 AC Stark effect, 742 Amplified Spontaneous Emission (ASE), 20, 46, 713 Angular momentum operators, acting on d states, 893 Annihilation operator, 135 Anomalous scattering factors, 334 Arago-Fresnel-Poisson bright spot, 412 Artificial atoms, 743 Atomic form factor, 291 Atomic orbitals, s, p and d, table, 511 Atomic scattering factor, 291 Atomic scattering factor, complex, 334 Atomic scattering length, complex, 334 Atomic structure factor, 291 Auger decay, 25 Autler–Townes effect, 742

B Babinet’s theorem, 414 Band model of solids, 554 Beam splitter, 233 Beer-Lambert law, 333 Beer-Lambert law, stimulated, 794 Beer-Lambert law, stimulated with polarization, 796 Bell’s inequalities, 13 Bending magnet, spectrum, 92 Bessel function, radial, 493 Biphotons, 232 Biphotons, entangled, 232, 234

Blackbody spectrum, 8 Bloch equations, analytical solutions, 708 Bloch equations, coherences, 703 Bloch equations, damping, 705 Bloch equations, dephasing, 703 Bloch equations, exact resonance, 711 Bloch equations, longitudinal relaxation, 704 Bloch equations, low intensity, 709 Bloch equations, occupations, 704 Bloch equations, optical, 701, 705 Bloch equations, power broadening, 718 Bloch equations, SASE pulse, 714 Bloch equations, steady state solution, 715 Bloch equations, transform limited pulse, 714 Bloch equations, transverse relaxation, 704 Bloch equations, upper state population, 711 Bloch-Rabi (BR) theory, 695 Bohr radius, 493 Bond specificity, 27 Born approximation, 302 Born-Oppenheimer approximation, 476 BR and KHD rates, link, 726 BR rates for Co L-shell, 728 BR rates, summary, 726 BR rates, time dependence at resonance, 731 BR theory, atom sum rule, 734 BR theory, for atomic sheet, 782 BR theory, link to KHD theory, 719 BR theory, model parameters, 706 BR theory, practical units, 707 BR theory, reduction to Einstein mode, 739 BR theory, saturation, 735 BR theory, transition rates definition, 705 BR theory, zero-point field, 725 BR time versus KHD band width, 721

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 J. Stöhr, The Nature of X-Rays and Their Interactions with Matter, Springer Tracts in Modern Physics 288, https://doi.org/10.1007/978-3-031-20744-0

911

912 Bragg diffraction, 484 Breit-Wigner cross section, 304 Brightness, 3, 19, 61 Brightness and coherence, link, 170 Brightness and partial coherence, 167 Brightness and photon density, 171 Brightness, average, 164 Brightness, concept, 160 Brightness, definition, 161 Brightness, higher order, 272 Brightness, peak, 166 Brightness, spectral, 163 Brilliance, 3, 61 Brilliance, see brightness, 161 C Casimir effect, 136 Causality, 306 Chaotic state, 900 Chaotic state, N -Photon Substate, 900 Chirality, 7, 130, 131, 551 Circular polarization, 127 Circular polarization, handedness, 128 Circular polarization, helicity, 128 Circular polarization, rotation in space, 128 Circular polarization, rotation in time, 128 Classical electron radius, 289 Clebsch-Gordon coefficients, 506, 533, 892, 894 Cloned biphotons, 818 Closure relation, 545, 896 Coherence, 168 Coherence and brightness, link, 170 Coherence and Fourier transform, 180 Coherence and uncertainty relations, 176 Coherence, concept, 176 Coherence cone, observer, 179 Coherence cone, source, 179 Coherence, definition, 176 Coherence, degree of second order, 263 Coherence factors, 170 Coherence factors, table, 172 Coherence, first order, 176 Coherence, first order definition, 195 Coherence, geometrical optics, 177 Coherence, higher order, 266 Coherence, link first and second order, 240 Coherence, partial, 168, 192 Coherence, partial lateral, 198 Coherence propagation, probability amplitudes, 219 Coherence propagation, quantum operators, 219

Index Coherence, QED order-dependent, 856 Coherence, second order, 908 Coherence, second order definition, 238 Coherence, second order of quantum states, 874 Coherence, second order spatial, 239 Coherence, space-time separability, 176 Coherence, spatial or lateral, 161 Coherence, temporal or longitudinal, 161 Coherence volume, 151 Coherence volume, brightness, 171 Coherence volume, per atom, 722 Coherence volume, per mode, 151, 722 Coherence volume, single mode, 491 Coherence volume, source, 171 Coherent diffractive imaging, 391 Coherent enhancement factor, 784 Coherent fraction, 168 Coherent state, 897 Coherent state, N -Photon Substate, 898 Coherent state, phase diffused, 899 Coincidence detection, 232 Collision broadening, 705 Compton scattering, 13, 17, 617 Compton scattering length, 291 Compton scattering, non-linear, 45 Compton wavelength, 85 Conductivity, AC, 321 Constraints in phasing, 430 Core hole clock lifetime, 605 Coulomb fields, 67 Coulomb gauge, 132, 133, 469 Creation operator, 135 Critical energy, 94 Cross section, definition, 288 Cross section, differential REXS/RIXS, 636 Cut-off energy, 83 Cyclotron frequency, 94 D Darwin-Prins theory, 351, 356 Darwin width, 356 De Broglie-Bohm Pilot Wave, 408 Degeneracy parameter, 173 Degeneracy parameter, synchrotron radiation, 173 Degeneracy parameter, XFEL, 173 Density matrix formalism, 702 Detection, light, 235 Detection, timescales, 237 Detectors, avalanche photodiodes, 237 Detectors, characteristics, 237 Detectors, charge coupled devices, 237

Index Detectors, superconducting single photon, 237 Detuning energy, 637 Dichroism, linear non-reciprocal, 364 Dichroism, magnetic, 362 Dichroism, magneto-chiral, 364 Dichroism, phenomenological treatment, 362 Dichroism, quantum formalism, 537 Dielectric displacement D, definition, 319 Dielectric function, 320 Dielectric polarization P, definition, 319 Dielectric response, 330 Dielectric response, frequency dependence, 320 Diffraction, 385 Diffraction imaging, polarization dependence, 415 Diffraction limit, 170, 175, 409 Diffraction limit, Abbe, 7 Diffraction limit, definition, 161 Diffraction limited source, 63, 104 Diffraction: operator formalism, 219 Diffraction pattern, first versus second order, 259 Diffraction: probability amplitude formalism, 219 Diffraction, QED description, 849 Diffraction, quantum theory, 250 Diffraction: wave formalism, 219 Diffraction, wave formulation, 395 Diffractive imaging, 385, 391 Dipolar transition width, polarization average, 698 Dipole approximation, 490 Dipole matrix element, angular, 508 Dipole matrix element, polarization dependence, 508, 895 Dipole matrix element, radial, 521 Dipole matrix elements, 895 Dipole matrix element, sum rules, 509 Dipole selection rules, 530 Dipole transition width, 503 Dipole transition width, partial occupation, 520 Dipole transition width, polarization average, 512 Dipole transition width, sum rule, 511 Dipole transitions, selection rules, 529 Dirac δ-function, 2D, 199 Dirac δ-function, 3D, 198 Dirac δ-function, definition, 884

913 Dirac δ-function, Gaussian approximation, 199 Dispersion corrections, atomic scattering, 294 Domains, antiferromagnetic, 32 Domains, ferromagnetic, 32 Doppler shift, 96 Double slit, first order patterns, 862 Double slit, plot of first order patterns, 866 Double slit, plot of second order patterns, 871 Double slit, second order patterns, 869 Drude conductivity, frequency dependence, 321 Dulong-Petit law, 774 Dynamic structure factor, 621 Dynamical structure factor, 622

E Einstein coefficients, 737 Einstein model, A and B coefficients, 738 Einstein model, absorption and emission, 736 Electret, 319 Electric dipole operator, 547, 549 Electric field, of relativistic point charge, 70 Electric permittivity, definition, 320 Electric quadrupole operator, 549 Electric quadrupole transitions, 547 Electric susceptibility, 320 Electro-magnetic field, electric component, 122 Electro-magnetic field, magnetic component, 122 Electro-magnetic field, polarization, 122 Electro-magnetic field, quantization, 132 Electro-magnetic fields, 67 Electro-magnetic wave equation, 121 Electromagnetic radiation, 87 Electromagnetic wave equation, 121 Electron bunch, Gaussian, 73 Electron bunch, ultrashort, 65 Electron, charge, 68 Electron coupling schemes, 533 Electronic temperature, definition, 772 Electron, relativistic energy, 64 Electron, relativistic mass, 64 Electron, rest mass, 64 EM field, normalization volume, 126 EM wave, energy, 122 EM wave, fluence, 122 EM wave, flux, 122

914 EM wave, intensity, 122 EM wave, momentum, 122 EM wave, Poynting vector, 122 EM waves, polarization, 127 Energy, EM wave, 122 Energy recovery linac, 63 Energy transfer, electronic system, 770 Entangled biphotons, 232 Entangled state, 900 Entanglement, 10 EPR paradox, 10 Ewald sphere, 442 EXAFS, 6 Exponential temporal decay, 183 Extinction coefficient, 323

F Faraday effect, 360 Faraday effect, magneto-optical, 7, 129 Far fields, 67 FERMI facility, 23 Fermi’s golden rule, 478 Ferroelectricity, 319 Field, relativistic observation cone, 71 Field superposition, coherent, 73 Field superposition, incoherent, 76 Fields, terahertz, 76 Fine structure constant, 291, 490 First Born approximation, 302 FLASH facility, 20 Florescence yield, 614 Fluctuation-dissipation theorem, 622 Fluence, classical definition, 122 Fluence, EM wave, 122 Fluence, quantum definition, 154 Flux, classical definition, 122 Flux, EM wave, 122 Flux, quantum definition, 154 Fock state, 139 Fourier limit, 272 Fourier theorem, diffraction, 404 Fourier transform, 1D, 885 Fourier transform, 2D, 888 Fourier transform, addition theorem, 432 Fourier transform, convolution theorem, 434 Fourier transform, definition, 884 Fourier Transform Holography (FTH), 431 Fourier transform, shift theorem, 432 Four wave mixing, 45 Fraunhofer approximation, 404 Fraunhofer diffraction, 404, 408 Fraunhofer diffraction formula, 203

Index Frequency spectrum, single electron, 92 Fresnel approximation, 403 Fresnel diffraction, 403 Fresnel length, 272 Friedel’s rule, 452

G Gauge, Coulomb, 132, 479 Gauge, Göppert-Mayer, 479 Gaussian, 883 Gaussian moment theorem, 240 Giant Magnetoresistance (GMR), 537, 577 Glauber states, 144, 216, 259 Grangier-Roger-Aspect experiment, 229

H Hanbury Brown–Twiss effect, 12 Hanbury Brown–Twiss experiment, 230 Harmonic oscillator, 134 Henke-Gullikson factors, 335 Henke-Gullikson formalism, 309 HERFD at Pt L-edge, 686 High Energy Density science (HED), 42 Higher coherence diffraction patterns, 267 Higher harmonic laser sources, 30 Hollow atoms, 42 Holography, 431 Holography, 2-slit reconstruction, 432, 434 Holography, Fourier transform, 431 Holography, in-line, 431 Holography, invention, 11 Holography, off-axis, 431 Holography, reference beam, 431 Hong-Ou-Mandel experiment, 232 Huygens–Fresnel principle, 7, 8, 337, 385, 397

I Impact parameter, 76 Index of refraction, 327 Inelastic electron scattering, 777 Inertia, 87 Inertial systems, 87 Infrared radiation, 78 Intensity, classical definition, 122 Intensity, EM wave, 122 Intensity, quantum definition, 154 Interaction Hamiltonian, second quantization, 744 Interference, 385 Irradiance, EM wave, 122

Index Isomorphous replacement, 442

J Jaynes-Cummings model, 698, 749 j j coupling, 533

K Kerr effect, 360 KHD band width versus BR time, 721 KHD, first order processes, 480 KHD formula, 478 KHD perturbation theory, 477 KHD theory, link to BR theory, 719 KHD theory, second order processes, 484 Kramers-Heisenberg-Dirac (KHD) theory, 478 Kramers-Kronig relations, 305

L Ladder operator, 134 Lamb shift, 11, 136 Lambert’s cosine law, 160 Lambertian source, 160 Laser, 12 Lateral coherence, measurement, 221 Lateral coherence, propagation, 186 Lattice temperature, 774 LCLS facility, 20 Length, proper, 63 Liénard - Wiechert equations, 66, 67 Liénard - Wiechert potentials, 17 Light detection, 235 Light, quantum states, 139 Light revolutions, 1 Linear absorption coefficient, 332 Linear polarization, 127 Liouville-von Neumann equation, 703 Lorentz contraction, 64 Lorentzian, 882 L S coupling, 533

M MAD, 25, 420, 442, 447 MAD, macromolecules, 448 MAD, non-crystalline samples, 452 Magnetic dipole operator, 549 Magnetic dipole transitions, 547 Magnetic field, of relativistic point charge, 70 Magnetic field response, 325

915 Magnetic permeability, definition, 319 Magnetic susceptibility χ, definition, 319 Magnetism, concepts, 552 Magnetization, Fe, Co, and Ni, 318 Magnetization M, definition, 318 Magneto-optical effect, 129 Magneto-optical Kerr Effect (MOKE), 360 Magnons, 675 Maser, 12 Maxwell-Bloch equations, 705 Maxwell-Bloch theory, 695 Maxwell’s equations, 120 MIR, 442 Mode, 135 Modes of light, 148 Molecular orbitals, 27 Molecular structure factor, 448 Mollow triplet, 749 Momentum, classical definition, 122 Momentum, EM wave, 122 Momentum, quantum definition, 154 Moseley’s law, 16 Mössbauer lifetime and width, 605 Mössbauer spectroscopy, 606 Multi-mode number state, 153 Multiple isomorphous replacement, 442 Multiple-wavelength anomalous diffraction, 448 Multiplet structure, 527 Mutual intensity, 194, 199, 200

N Natural linewidth, 503 Natural linewidth, definition, 499 NEXAFS, 6, 26 NMR, 695 NMR, relaxation times T1 and T2 , 705 No-cloning theorem, 647, 875 Non-linear diffraction, polarization dependence, 825 Non-linear diffraction, theory versus experiment, 830 Non-linear effects, definition, 45 Non-linear film transmission, 785 Non-linear film transmission sum rule, 793 Non-linear transmission, atom to film, 792 Non-linear transmission through transition metals, 798 Non-linear transmission versus diffraction, 815 Non-resonant inelastic x-ray scattering, see XRS, 623

916 NOON quantum state, 861 NOON state, 900 Normalization volumes, x-rays, 725 Normal ordering, 195 Normal ordering of operators, 212 Number states, 134 Number state, N -Photon, 901 Numerical aperture, 409, 412 Nyquist criterion, 444 O Obliquity factor, 161, 400 Operator, electric field, 138 Operator, magnetic field, 138 Operators, angular momentum, 893 Operators, spherical tensors, table, 890 Operators, spin, 893 Operator, vector potential, 138 Optical activity, 129 Optical Bloch equations, 696 Optical “constants”, see optical parameters, 323 Optical parameters, 327 Optical parameters, relation to permittivity and permeability, 323 Optical pumping, 11 Optical theorem, 305, 343 Optical theorem, energy conservation, 732 Optical theorem, sum rule, 733 Orbital magnetic moments, 556 Orbital magnetism, 562 Orbitals, s, p, and d , 891 Orientation factors, 366 Orientational order, 364, 417 Oscillator strength, definition, 525 Oversampling, 446 P Parametric down conversion, 232 Paraxial approximation, 178, 402 Parity, 129, 550 Parseval’s theorem, 171 Parseval’s theorem, 1D, 885 Parseval’s theorem, 2D, 888 Partial coherence, 192 Partial coherence, quantum description, 192 Patterson map, 430 Pauli equation, 468 Peak brightness, master equation, 171 PEEM, 389 Permeability, definition, 319 Permeability, frequency dependence, 321

Index Permittivity, definition, 320 Permittivity, frequency dependence, 320 Perturbation theory, Kramers-HeisenbergDirac (KHD), 477 Perturbation theory, time dependent, 474 Phase contrast, 459 Phase problem, 428 Phase space, 164 Phase space, Liouville’s theorem, 164 Phase space volume, 164 Phase velocity, EM wave, 122 Phasing constraints, 430 Photoelectric effect, 9 Photoemission Electron Microscopy (PEEM), 387 Photoemission spectroscopy, 6 Photoemission spectrum, core levels, 497 Photon antibunching, 697, 743, 761 Photon bunching, 230, 761 Photon, coherence volume, 148 Photon degeneracy parameter, 3, 173 Photon degeneracy parameter, definition, 140 Photon, density of states, 148 Photon, existence proof, 229 Photon flux, areal, 164 Photon flux, quantum definition, 491 Photon flux, spectral, 164 Photon-matter interaction Hamiltonian, 468 Photon, modes, 135, 148 Photon modes, number per energy, 150 Photon modes, number per volume, 151 Photon spin, 128 Photon, virtual, 80 Pilot wave theory, 408 Polarization, circular, 127, 131 Polarization, degree of, 131 Polarization, elliptical, 131 Polarization, linear, 127, 131 Polarization, synchrotron radiation, 105 Polarization, use in microscopy, 391 Polarization vectors, definition, 130 Power broadening, 718, 719 Power conservation, two-photon diffraction, 253 Poynting vector, 122 Probability amplitudes, 406 Propagation of first order coherence, 186 Propagation of second order coherence, 241 Protein crystallography, growth, 24 Ptychography, 446 Pulses, transform limited, 114 Pulse, temporal flat-top, 181

Index Pulse, temporal Gaussian, 183 Pulse, THz, 78 Q Quadrupole tensor, 545 Quadrupole tensor, matrix elements, 545 Quantum diffraction, 211, 250 Quantum diffraction, evolution from first to second order, 875 Quantum diffraction, first order, 855 Quantum diffraction, formulation, 853 Quantum diffraction, reduction to wave formalism, 868 Quantum diffraction, second order, 855 Quantum Electrodynamics (QED), 3, 11 Quantum information science, 13 Quantum mechanics, formulations, 5 Quantum optics, 3, 12, 227 Quantum regression theorem, 746 Quantum states, coherent, 259 Quantum states, collective 2-mode, 857 Quantum states, collective chaotic, 859 Quantum states, collective chaotic substates, 859 Quantum states, collective coherent, 259, 858 Quantum states, collective coherent substates, 858 Quantum states, collective phase-diffused, 858 Quantum states, entangled states, 861 Quantum states of light, 139 Quantum states of light, diffraction patterns, 897 Quantum states of light, formalism, 856 Quantum states of light, generation, 851 Quantum states, multi-mode, 153 Quantum states, N -Photon Number State, 861 Quantum states, plot of substate distributions, 860 Quantum states, properties, 140 Quantum states, single-mode chaotic, 146 Quantum states, single-mode coherent, 143 Quantum states, single-mode number, 139 Quantum states, summary, 861 Quasi-classical states, 144 Quasi-homogeneous, 166, 196 Quasi-stationary, 196 R Rabi energy, 699

917 Rabi energy width, 699 Rabi frequency, 698 Rabi frequency, x-rays, 699 Racah’s spherical tensors, table, 890 RADAR, 695 Radar, 12 Radial dipole matrix element, 521 Radial wavefunctions, atom, 492 Radiation, 87 Radiation field, Hamiltonian, 136 Radiation, angular pattern, 88 Radiation, bremsstrahlung, 80 ˇ Radiation, Cerenkov, 80 Radiation, dipole, 90 Radiation, synchrotron, 80, 86 Radiation, terahertz, 81 Radiation, transition, 80 Radiation, x-rays, 83 Raman scattering, 635 Rayleigh criterion, 412 Rayleigh scattering, 635 Rayleigh scattering, classical, 298 Rayleigh-Sommerfeld diffraction theory, 400 Reciprocal lattice, 442 Refractive index, 327 Refractive index, complex, 323 Refractive index, real, 323 Relativistic concepts, 63 Relativistic time, 64 Resolution, diffraction limited, 409 Resonance fluorescence, 697, 742 Resonance fluorescence spectrum, 749 Resonance fluorescence, bunching and antibunching, 759 Resonance fluorescence, coherent and incoherent parts, 753 Resonance fluorescence, dephasing, 758 Resonance fluorescence, first order coherence, 745 Resonance fluorescence, Rabi splitting, 749 Resonance fluorescence, second order coherence, 756 Resonant processes, classical description, 293 Resonant scattering, 635 Resonant scattering, classical, 296 Resonant scattering, KHD formulation, 632 Resonant scattering, REXS and RIXS, 632 Resonant scattering, spontaneous, 484 Resonant scattering, stimulated, 485 Rest frame, 63 Retardation effects, 66

918 REXS, 6 REXS, classical versus quantum, 647 REXS, cross section, 641 REXS, forward scattering, 647 REXS, intermediate state interference, 648 REXS, K-edge of N2 and O2 , 653 REXS, polarization dependence, 656 REXS, role of instrumental resolution, 642 REXS, scattering time, 651 REXS, spin dependence, 656 REXS, spontaneous versus stimulated, 643 REXS, thin film case, 661 REXS, vibrational finestruture, 653 REXS/RIXS, configuration formalism, 636 REXS/RIXS, differential cross section, 636 REXS/RIXS, double matrix element, 633 REXS/RIXS, one electron formalism, 636 REXS/RIXS, terminologies, 639 RIXS, 6 RIXS and reduced linewidth XAS (HERFD), 685 RIXS, differential cross section, 663 RIXS, direct and indirect, 640 RIXS, integrated cross section, 665 RIXS, magnon dispersion, 675 RIXS of chemisorbed molecules, 681 RIXS of Glycine on Cu(110), 682 RIXS of K-edge in N2 and O2 , 668 RIXS of L-edge in transition metal oxides, 673 RIXS of L-edge in transition metals, 676 RIXS, one-step or coherent, 640 RIXS, polarization dependence, 682 RIXS reduction to XES, 668 RIXS, role of instrumental resolution, 666 RIXS, stimulated, 834 RIXS, two-step, 664 RIXS, two-step or incoherent, 640 Röntgen rays, 14 Rotating wave approximation, 634, 699, 703 Rydberg resonances, 513

S Sample delivery, 37 SASE, gain length, 110 Saturation of transitions, 735 Saupe matrix, 366 Scanning transmission x-ray microscopy (STXM), 32, 387 Scattering length, definition, 288 Scattering, dynamical, 351 Scattering, kinematical, 351

Index Schell model source, 202 Search light effect, 363, 541 Second harmonic generation, 45 Second order coherence, 908 Second quantization, 744 Self Amplified Spontaneous Emission (SASE), 20, 63, 109 Self-induced transparency, SIT, 47 Semi-classical, 3 SI units, Aantities, 881 Siegert relation, 240 Single-mode number state, 139 Single-mode states, properties, 140 Single-wavelength Anomalous Diffraction (SAD), 25, 442 Skin depth, 325 SLAC, 65 Small angle x-ray scattering, 391 Source, bending magnet, 97 Source, undulator, 97 Space-time separability, 161 Speckle pattern, 391 Spectral brightness, 19 Spectro-microscopy, 27, 387, 389 Spectroscopy, polarization dependence, 28 Spherical harmonics, 504, 890 Spherical tensors, 890 Spherical tensors, Racah, 504 Spin accumulation, 578 Spin Hall effect, 537, 581 Spin operators, acting on spin states., 893 Spin orbitals, atom, 492 Spin-orbit coupling, Hamiltonian, 470 Spin-orbit functions, p states, 506 Spin-orbit functions, s, p, and d states, 892 Spin-orbit Hamiltonian, matrix elements, 568 Spin-orbit interaction, 556, 571 Spin-orbit splitting, 505 Spinor formalism, 570 Spontaneous emission, 46 Spontaneous parametric down conversion, 45, 232 Spontaneous versus stimulated diffraction pattern, 820 Stark effect, dynamical, 742 Statistical optics, 11, 12, 227 Stimulated emission, 45, 46 Stimulated response, single atom versus collective, 802 Stimulated REXS/RIXS, interplay, 840 Stimulated RIXS gain, 842

Index Stimulation, dependence on pulse coherence time, 800 Stimulation, mode dependent, 822 Stimulation, soliton model, 822 Stoner model, 554 Storage rings, 18 Storage rings, ultimate, 62, 104 Structure factor, 448 STXM, 387, 389 Sudden approximation, 476 Sum rule, atomic form factor, 314 Sum rule, Thomas-Reiche-Kuhn, 313 Symmetry, inversion, 129 Symmetry, parity, 129 Symmetry, time reversal, 129 Synchrotron radiation, 86 Synchrotron radiation, generations, 17 Synchrotron radiation, history, 17 Synchrotron radiation, lateral coherence, 101 Synchrotron radiation, polarization, 105 Synchrotron radiation, spectrum, 92

T Tensor order parameter, 366 Terahertz radiation, 78 Thomas-Reiche-Kuhn sum rule, 313 Thomson scattering, 25, 484 Thomson scattering, atom, 291 Thomson scattering, classical, 284 Thomson scattering, differential cross section, 289, 620 Thomson scattering, inelastic, 622 Thomson scattering length, 289 Thomson scattering, phase shifts, 288 Thomson scattering, quantum formulation, 618 Thomson scattering, single electron, 284 Thomson scattering, single spin, 284 Thomson scattering, spins, 291, 472 Three-temperature model, 769 THz, 76 THz pulse, half cycle, 79 Time-bandwidth product, 184 Time, proper, 63 Time, relativistic, 64 Time, retarded, 66 Tomography, 387 Transform limit, 170, 175 Transform limit, definition, 161 Transient spin phenomena, 577 Transition matrix element, angular, 504

919 Transition matrix element, factorization, 504 Transition matrix element, radial, 504, 521 Transition matrix element sum rules, 895 Transition probabilities, p → d, 895 Transition probability, definition, 478 Transition state, 38 Transmission x-ray microscopy (TXM), 387, 389 Triggering of processes, 38 Two-level atom, 701 Two-level atom, BR, KHD and Einstein theories, 736 Two-level system, 697 2-photon absorption, 486 Two-photon chaotic state, 249 Two-photon coherent state, 246 2-photon diffraction, 244 2-photon diffraction patterns, 251 Two-photon entangled state, 248 Two-photon interference, 232 Two-photon number state, 248 Two-temperature model, 769, 774

U Undulator, angular distribution, 99 Undulator, frequency spectrum, 96 Undulator source, 96 Units and values of important quantities, 881 UPS, 25

V Vacuum field, 136 Valence electron excitations, 777 Van Cittert-Zernike formula, 205 Van Cittert-Zernike theorem, 11, 171, 197 Van Cittert-Zernike theorem, quantum optics derivation, 210 Van Cittert-Zernike theorem, statistical optics derivation, 199 Vector potential, 132 Velocity fields, 67, 68 Virtual photon exchange, 80 Virtual photons, 80, 136 Visibility, 222 Voigt lineshape, 883 Vortex, magnetic, 32

W Water, absorption spectrum, 26 Wave equation, 121 Wave-particle duality, 9, 405

920 Wave propagation, 186 Weisskopf-Wigner linewidth, 503 Weizsäcker-Williams method, 61, 79 Wiener-Khintchine theorem, 11, 171, 197 Wigner 3 j coefficients, 506 Wigner coefficients, 533 Wigner-Eckart theorem, 894

X X-PEEM, 387 X-ray ablation threshold, 775 X-ray absorption, coefficient, 332 X-ray absorption edges, labels, 309 X-ray absorption, law, 330 X-ray absorption spectroscopy, history, 15 X-ray beam parameters, flux, fluence, intensity, 768 X-ray crystallography, 17 X-ray crystallography, history, 442 X-ray damage, definition, 767 X-ray damage, temporal evolution, 768 X-ray emission spectroscopy (XES), 30, 596 X-ray emission spectroscopy, history, 15 X-ray fluorescence yield, 614 X-ray free electron lasers, history, 19 X-ray free-electron laser, concept, 108 X-ray imaging, speckle pattern, 391 X-ray interactions, relative size, 472 X-ray magnetic circular dichroism (XMCD), 30, 362 X-ray magnetic linear dichroism (XMLD), 29, 362 X-ray natural circular dichroism (XNCD), 30, 362 X-ray natural linear dichroism (XNLD), 29, 362 X-ray Nobel prizes, 15 X-ray photoemission electron microscopy (X-PEEM), 387 X-ray Raman scattering (XRS), 623 X-ray resonances, importance, 27 X-ray response, E versus H, 325 X-ray tomography, 387 X-ray transparency, 43 X-ray transparency, introduction, 780 X-ray transparency, non-resonant, 809 X-ray transparency, non-resonant versus resonant, 803 X-ray transparency, resonant, 804 X-ray transparency, self-induced, 797 X-rays absorption (XAS), 489 X-rays, field strength, 125

Index XANES, 6, 26 XAS, configuration model, 505 XAS, cross section, 501 XAS cross section, non-resonant, 494 XAS cross section, partial occupation, 521 XAS, dipole matrix element, 504 XAS, KHD formulation, 489 XAS, L-shell radial matrix element, 521 XAS, lineshape, 501 XAS lineashapes, theory versus experiment, 523 XAS lineshapes, atoms versus solids, 523 XAS linewidth, 499, 501 XAS linewidth, Auger contribution, 499 XAS linewidth, radiative contribution, 499 XAS, M-shell radial matrix element, 521 XAS, multiplet structure, 527 XAS, N2 molecule, 515 XAS, Ne atoms, 513 XAS, non-resonant, 492 XAS, O2 molecule, 515 XAS, one-electron model, 505 XAS, oscillator strength, 525 XAS, polarization averaged sum rules, 526 XAS, polarization dependence, 508 XAS, resonant, 499 XAS, spontaneous cross section, 701 XAS, 3d transition metals, 517 XES, Auger linewidth contribution, 602 XES, decay time, 599 XES, definition, 596 XES, dipolar transition width, 604 XES, dipole matrix element, 597, 604 XES, emission rate, 604 XES, KHD formulation, 597 XES, K-shell in N2 molecule, 609 XES, line width, 599 XES, L-shell in 3d metals, 611 XES of Ne atoms, 607 XES, radial dipole matrix element, 614 XES, role of zero-point field, 597 XES, spontaneous, 597 XES, stimulated, 597 XFEL concept, 108 XFEL, history, 19, 23 XFEL, pulses, 112 XFEL, SASE, 109 XFEL, spatial coherence, 109 XFEL, temporal coherence, 109 XMCD, Cu-Phthalocyanine, 566 XMCD effect, 375 XMCD, orbital moment sum rule, 559 XMCD, quantum formulation, 552

Index XMCD, spin moment sum rule, 564 XMCD, test of sum rules, 566 XMLD, Cu-Phthalocyanine, 585 XMLD effect, origin, 582 XMLD, multiplet enhancement, 588 XMLD, transition metals, 586 XNCD, quantum formulation, 547 XNCD, two types, 550 XNLD, applications, 546 XNLD, quadrupolar formulation, 543 XNLD, quantum formulation, 540 XPS, 25 XRS, Compton background, 626

921 XRS versus XAS cross section, 626 XRS, momentum transfer, 625

Y Young’ double slit, quantum formulation, 849 Young’s experiment, 12, 849

Z Zero-point energy, 9, 135 Zero-point field, 136, 725